a “magic” approach to octonionic
Rosenfeld spaces

12 December 2022

Alessio Marrania, Daniele Corradettib, David Chesterc,

Raymond Aschheimc, Klee Irwinc

_$a$ Instituto de Física Teorica, Departamento de Física,_

_Universidad de Murcia, Campus de Espinardo, E-30100, Spain_

_email:_<EMAIL_ADDRESS>
_$b$ Universidade do Algarve, Departamento de Matemática,_

_Campus de Gambelas, 8005-139 Faro, Portugal_

_email:_<EMAIL_ADDRESS>
_$c$ Quantum Gravity Research,_

_Topanga Canyon Rd 101 S., California CA 90290, USA_

_email:_<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

In his study on the geometry of Lie groups, Rosenfeld postulated a strict
relation between all real forms of exceptional Lie groups and the isometries
of projective and hyperbolic spaces over the (rank-2) tensor product of
Hurwitz algebras taken with appropriate conjugations. Unfortunately, the
procedure carried out by Rosenfeld was not rigorous, since many of the
theorems he had been using do not actually hold true in the case of algebras
that are not alternative nor power-associative. A more rigorous approach to
the definition of all the planes presented more than thirty years ago by
Rosenfeld in terms of their isometry group, can be considered within the
theory of coset manifolds, which we exploit in this work, by making use of all
real forms of Magic Squares of order three and two over Hurwitz normed
division algebras and their split versions. Within our analysis, we find 7
pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any
interpretation within Rosenfeld’s framework. We carry out a similar analysis
for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian
symmetric cosets which do not have any interpretation à la Rosenfeld.

###### Contents

  1. 1 Introduction
  2. 2 Real forms of Magic Squares of order 3 and 2
    1. 2.1 Hurwitz algebras and their split versions
    2. 2.2 Trialities and derivations
    3. 2.3 Euclidean and Lorentzian simple rank-3 Jordan algebras
    4. 2.4 Real forms of the Freudenthal-Tits Magic Square
    5. 2.5 Magic Square of order 2
    6. 2.6 Octonionic entries of Magic Squares of order 2 and 3
  3. 3 “Magic” formulæ for Rosenfeld planes
    1. 3.1 Coset manifolds
    2. 3.2 “Magic” formulæ
  4. 4 Octonionic Rosenfeld planes
    1. 4.1 $\mathbb{O}\simeq\mathbb{R}\otimes\mathbb{O}$
    2. 4.2 $\mathbb{C}\otimes\mathbb{O}$
    3. 4.3 $\mathbb{H}\otimes\mathbb{O}$
    4. 4.4 $\mathbb{O}\otimes\mathbb{O}$
  5. 5 “Magic” formulæ for Rosenfeld lines
  6. 6 Octonionic Rosenfeld lines
    1. 6.1 $\mathbb{O}\simeq\mathbb{R}\otimes\mathbb{O}$
    2. 6.2 $\mathbb{C}\otimes\mathbb{O}$
    3. 6.3 $\mathbb{H}\otimes\mathbb{O}$
    4. 6.4 $\mathbb{O}\otimes\mathbb{O}$
  7. 7 Conclusions

##  1 Introduction

Around the half of the XX century, geometric investigations on the octonionic
plane gave rise to a fruitful mathematical activity, culminating into the
formulation of Tits-Freudenthal Magic Square [Fr65, Ti]. The Magic Square is
an array of Lie algebras, whose entries are obtained from two Hurwitz
algebras111Even though the original construction by Tits was not symmetric in
the two entries (cf. the third of (1)), the square turned out to be symmetric
and was, therefore, dubbed as “magic”. and enjoy multiple geometric and
algebraic interpretations [BS00, BS03, SH, Sa, El02, El04], as well as
physical applications [GST, DHW, CCM].

The entries $\mathfrak{m}_{3}(\mathbb{A}_{1},\mathbb{A}_{2})$ of the Tits-
Freudenthal Magic Square can be defined equivalently [BS03] using the Tits
[Ti], Barton-Sudbery [BS03] (also cf. [Ev]), and Vinberg [Vin66]
constructions222The ternary algebra approach of [Ka] was generalised by Bars
and Günaydin in [BG] to include super Lie algebras. Generalizations to affine,
hyperbolic and further extensions of Lie algebras have been considered in
[Pa].,

$\mathfrak{m}_{3}(\mathbb{A}_{1},\mathbb{A}_{2})=\left\\{\begin{array}[]{l}\mathfrak{der}(\mathbb{A}_{1})\oplus\mathfrak{der}(\mathfrak{J}_{3}(\mathbb{A}_{2}))\,\dot{+}\,\text{Im}\mathbb{A}_{1}\otimes\mathfrak{J}_{3}^{\prime}(\mathbb{A}_{2});\\\\[5.0pt]
\nobreak\leavevmode\\\
\mathfrak{der}(\mathbb{A}_{1})\oplus\mathfrak{der}(\mathbb{A}_{2})\,\dot{+}\,\mathfrak{sa}(3,\mathbb{A}_{1}\otimes\mathbb{A}_{2});\\\
\nobreak\leavevmode\\\
\mathfrak{tri}(\mathbb{A}_{1})\oplus\mathfrak{tri}(\mathbb{A}_{2})\,\dot{+}\,3(\mathbb{A}_{1}\otimes\mathbb{A}_{2}),\end{array}\right.$
(1)

where $\mathbb{A}_{1}$ and $\mathbb{A}_{2}$ are normed Hurwitz division
algebras $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ or the split
counterparts $\mathbb{C}_{s},\mathbb{H}_{s},\mathbb{O}_{s}$. Here,
$\mathfrak{J}_{3}(\mathbb{A})$ denotes the Jordan algebra of $3\times 3$
Hermitian matrices over $\mathbb{A}$ and
$\mathfrak{J}_{3}^{\prime}(\mathbb{A})$ its subspace of traceless elements.
The space of anti-Hermitian traceless $n\times n$ matrices over
$\mathbb{A}_{1}\otimes\mathbb{A}_{2}$ is denoted
$\mathfrak{sa}(n,\mathbb{A}_{1}\otimes\mathbb{A}_{2})$. Details of the
commutators and the isomorphisms between these Lie algebras can be found e.g.
in [BS03]. The choice of division or split $\mathbb{A}_{1},\mathbb{A}_{2}$
yields different real forms : for the Tits construction, further possibilities
are given by allowing the Jordan algebra to be Lorentzian, denoted
$\mathfrak{J}_{2,1}$, as described in [CCM] (see Sec. 2.3 below);
equivalently, one can introduce overall signs in the definition of the
commutators between distinct components in the Vinberg or Barton-Sudbery
constructions, as described in [ABDHN]. For a complete listing of all
possibilities333Concerning the extension of Freudenthal-Tits Magic Square to
various types of “intermediate” algebras, the extension to the sextonions
$\mathbb{S}$ (between $\mathbb{H}$ and $\mathbb{O}$) is due to Westbury [Wes],
and later developed by Landsberg and Manivel [LM01, LM06], whereas further
extension to the tritonions $\mathbb{T}$ (between $\mathbb{C}$ and
$\mathbb{H}$) was discussed by Borsten and one of the present authors in
[BM]., see [CCM]. Recently [WDM], Wilson, Dray and Manogue gave a new
construction of the Lie algebra $\mathfrak{e}_{8}$ in terms of $3\times 3$
matrices such that the Lie bracket has a natural description as matrix
commutator; this led to a new interpretation of the Freudenthal-Tits Magic
Square of Lie algebras444Magic Squares of order 2 of Lie groups have been
investigated in [DHK]. Interestingly, the Freudenthal-Tits Magic Square of Lie
groups was observed to be non-symmetric since [Yo85]., acting on themselves by
commutation.

While Tits was more interested in algebraic aspects, Freudenthal interpreted
every row of the Magic Square in terms of a different kind of geometry, i.e.
elliptic, projective, symplectic and meta-symplectic for the first, second,
third and fourth row, respectively [LM01]. This means that, while the algebras
taken in account were always the Hurwitz algebras
$\mathbb{R},\mathbb{C},\mathbb{H}$ and $\mathbb{O}$, all Lie groups were
arising from considering different type of transformations such as isometries,
collineations and homographies of the plane [Fr65]. On the other hand,
Rosenfeld conceived every entry of the $4\times 4$ array of the Magic Square
as the Lie algebra of the (global) isometry group of a “generalized”
projective plane, later called Rosenfeld plane [Ro98, Ro97]. Thus, while in
Freudenthal’s framework the projective plane was always the same but,
depending on the considered row of the Magic Square, the kind of
transformations was changed; in Rosenfeld’s picture the group of
transformations was always the isometry group, and different Lie groups
appeared considering different planes over tensor product of Hurwitz algebras.

To be more precise, since the exceptional Lie group $\text{F}_{4}$ is the
isometry group of the octonionic projective plane $\mathbb{O}P^{2}$, then
Rosenfeld regarded $\text{E}_{6}$ as the isometry group of the bioctonionic
“projective” plane $\mathbb{\left(C\otimes\mathbb{O}\right)}P^{2}$,
$\text{E}_{7}$ as the isometry group of the quaternoctonionic “projective”
plane $\mathbb{\left(H\otimes\mathbb{O}\right)}P^{2}$, and $\text{E}_{8}$ as
the isometry group of the octooctonionic “projective” plane
$\mathbb{\left(O\otimes\mathbb{O}\right)}P^{2}$. Despite Rosenfeld’s
suggestive interpretation, it was soon realized that the planes identified by
Rosenfeld did not satisfy projective axioms, and this explains the quotes in
the term “projective”, which would have to be intended only in a vague sense.
It was the work of Atsuyama, followed by Landsberg and Manivel, to give a
rigorous description of the geometry arising from Rosenfeld’s approach [Atsu2,
LM01, AB]. Indeed (unlike $\mathbb{O\simeq R\otimes O}$) $\mathbb{C\otimes
O}$, $\mathbb{H\otimes O}$ and $\mathbb{O\otimes O}$ are not division
algebras, thus preventing a direct projective construction; moreover (unlike
$\mathbb{C\otimes O}$), Hermitian $3\times 3$ matrices over $\mathbb{H\otimes
O}$ or $\mathbb{O\otimes O}$ do not form a simple Jordan algebra, so the usual
identification of points (lines) with trace 1 (2) projection operators cannot
be made [Ba]. Nonetheless, they are in fact geometric spaces, generalising
projective spaces, known as “buildings”, on which (the various real forms of)
exceptional Lie groups act as isometries. Buildings where originally
introduced by Tits in order to provide a geometric approach to simple Lie
groups, in particular the exceptional cases, but have since had far reaching
implications; see, for instance, [Ti2, Te] and Refs. therein.

More specifically, Rosenfeld noticed that, over $\mathbb{R}$, the isometry
group of the octonionic projective plane was the compact form of
$\text{F}_{4}$, i.e. $\text{F}_{4\left(-52\right)}$, while the other, non-
compact real forms, i.e. the split form $\text{F}_{4\left(4\right)}$ and
$\text{F}_{4\left(-20\right)}$, were obtained as isometry groups of the split-
octonionic projective plane $\mathbb{O}_{s}P^{2}$ and of the octonionic
hyperbolic plane $\mathbb{O}H^{2}$ [CMCAb, CMCAa], respectively. Thence, he
proceeded in relating all real forms of exceptional Lie groups with projective
and hyperbolic planes over tensorial products of Hurwitz algebras [Ro97,
Ro98].

Despite being very insightful, to the best of our knowledge Rosenfeld’s
approach was never formulated in a systematic way, and a large part of the
very Rosenfeld planes were never really rigorously defined. In the present
investigation, we give a systematic and explicit definition of all
octonionic555The quaternionic, complex and real “Rosenfeld planes” and
“Rosenfeld lines” can be obtained as proper sub-manifolds of their octonionic
counterparts, so we will here focus only on the latter ones. “Rosenfeld
planes” as coset manifolds, by exploiting all real forms of Magic Squares of
order 3 over Hurwitz algebras and their split versions, thus all possible real
forms of Freudenthal-Tits Magic Square. In doing so, we will also find a total
of 10 coset manifolds, namely 7 “planes” and 3 “lines”, whose definition is
straightforward according to our procedure, but that apparently do not have
any interpretation according to Rosenfeld’s approach.

The plan of the paper is as follows. In Sec. 2 we will resume all the
algebraic machinery used in this paper, i.e. Hurwitz algebras, their triality
and derivation symmetries, Euclidean and Lorentzian cubic Jordan algebras,
Magic Squares of order 3 and 2, etc. In Sec. 3 we will then present the
“magic” formulæ which we will use in order to define the octonionic Rosenfeld
planes in Sec. 4; then, we will do the same for octonionic Rosenfled lines in
Secs. 5 and 6. Some final remarks and an outlook are given in Sec. 7.

##  2 Real forms of Magic Squares of order 3 and 2

###  2.1 Hurwitz algebras and their split versions

Let the octonions $\mathbb{O}$ be the only unital, non-associative, normed
division algebra with $\mathbb{R}^{8}$ decomposition of $x\in\mathbb{O}$ given
by

$x=\sum_{k=0}^{7}x_{k}\text{i}_{k},$ (2)

where $\left\\{\text{i}_{0}=1,\text{i}_{1},...,\text{i}_{7}\right\\}$ is a
basis of $\mathbb{R}^{8}$ and the multiplication rules are mnemonically
encoded in the Fano plane (see left side of Fig. 1), along with
$\text{i}_{k}^{2}=-1$ for $k=1,...,7$. Let the norm $N\left(x\right)$ be
defined as

$N\left(x\right):=\sum_{k=0}^{7}\left(x_{k}\text{i}_{k}\right)^{2},$ (3)

and its polarisation as $\left\langle
x,y\right\rangle:=N\left(x+y\right)-N\left(x\right)-N\left(y\right).$ Then the
real part of $x$ is $\mathfrak{R}\left(x\right):=\left\langle
x,1\right\rangle$ and $\overline{x}:=2\left\langle x,1\right\rangle-x.$ We
then have that $N\left(x\right)=\overline{x}x,\,\,\,\left\langle
x,y\right\rangle=\overline{x}y+\overline{y}x,$and that

$N\left(xy\right)=N\left(x\right)N\left(y\right),$ (4)

or, in other words, that octonions are a composition algebra with respect to
the Norm $N$ defined by (3).

We now define as the algebra of quaternions $\mathbb{H}$ as the subalgebra of
$\mathbb{O}$ generated by the elements
$\left\\{\text{i}_{0}=1,\text{i}_{1},\text{i}_{2},\text{i}_{3}\right\\}$ and
the algebra of complex numbers $\mathbb{C}$ as the subalgebra of $\mathbb{O}$
generated by the elements $\left\\{\text{i}_{0}=1,\text{i}_{1}\right\\}$.

Figure 1:  Multiplication rule of octonions $\mathbb{O}$ (_left_) and of
split-octonions $\mathbb{O}_{s}$ (_right_) as real vector space
$\mathbb{R}^{8}$ in the basis
$\left\\{\text{i}_{0}=1,\text{i}_{1},...,\text{i}_{7}\right\\}$. In the case
of the octonions $\text{i}_{0}^{2}=1$ and $\text{i}_{k}^{2}=-1$ for $k=1...7$,
while in the case of split-octonions $\text{i}_{k}^{2}=+1$, for $k=1,2,3$ and
$\text{i}_{k}^{2}=-1$ for $k\neq 1,2,3$.

Thus, the split-octonions $\mathbb{O}_{s}$ can be defined as the only unital,
non-associative, normed algebra with $\mathbb{R}^{8}$ decomposition of
$x\in\mathbb{O}_{s}$ given by (2) where, again,
$\left\\{\text{i}_{0}=1,\text{i}_{1},...,\text{i}_{7}\right\\}$ is a basis of
$\mathbb{R}^{8}$, but the multiplication rules are encoded in a different
variant of the Fano plane, given by the right side of Fig. 1, along with
$\text{i}_{k}^{2}=1$ for $k=1,...,3$ and $\text{i}_{k}^{2}=-1$ for $k=4,5,6,7$
. Norm and conjugation are defined as in the octonionic case and, as for the
division octonions, we obtain that split octonions are a composition algebra
with respect to the Norm $N$ defined by (3). We then define the algebra of
split-quaternions $\mathbb{H}_{s}$ as the subalgebra of $\mathbb{O}_{s}$
generated by
$\left\\{\text{i}_{0}=1,\text{i}_{1},\text{i}_{4},\text{i}_{5}\right\\}$ and
the split-complex (or hypercomplex) numbers $\mathbb{C}_{s}$ as the subalgebra
of $\mathbb{O}_{s}$ generated by
$\left\\{\text{i}_{0}=1,\text{i}_{4}\right\\}$.

###  2.2 Trialities and derivations

Let $\mathbb{A}$ be an algebra and $\text{End}\left(\mathbb{A}\right)$ the
associative algebra of its linear endomorphisms. Then, the triality algebra
$\mathfrak{tri}\left(\mathbb{A}\right)$ is the Lie subalgebra of
$\mathfrak{so}\left(\mathbb{A}\right)\oplus\mathfrak{so}\left(\mathbb{A}\right)\oplus\mathfrak{so}\left(\mathbb{A}\right)$
defined as

$\mathfrak{tri}\left(\mathbb{A}\right):=\left\\{\left(A,B,C\right)\in\text{End}\left(\mathbb{A}\right)^{\oplus
3}:A\left(xy\right)=B\left(x\right)y+xC\left(y\right),\forall
x,y\in\mathbb{A}\right\\},$ (5)

where the Lie bracket are those inherited as a subalgebra. Derivations are a
special case of trialities of the form $\left(A,A,A\right)$ and therefore they
again form a Lie algebra, defined as

$\mathfrak{der}\left(\mathbb{A}\right):=\left\\{A\in\text{End}\left(\mathbb{A}\right):A\left(xy\right)=A\left(x\right)y+xA\left(y\right),\forall
x,y\in\mathbb{A}\right\\},$ (6)

where the bracket is given by the commutator. In the case of
$\mathbb{A}=\mathbb{C},\mathbb{H},\mathbb{O}$ we intended
$\mathfrak{so}\left(\mathbb{A}\right)$ as $\mathfrak{so}\left(2\right)$,
$\mathfrak{so}\left(4\right)$ and $\mathfrak{so}\left(8\right)$ respectively;
while, for their split companions, we intended
$\mathfrak{so}\left(\mathbb{A}_{s}\right)$ as $\mathfrak{so}\left(1,1\right)$,
$\mathfrak{so}\left(2,2\right)$ and $\mathfrak{so}\left(4,4\right)$. Triality
and derivation Lie algebras over Hurwitz algebras are summarized in Table 1.

| $\mathbb{R}$ | $\mathbb{C}$ | $\mathbb{H}$ | $\mathbb{O}$
---|---|---|---|---
$\mathbb{\mathfrak{tri}\left(\mathbb{A}\right)}$ | Ø | $\mathfrak{u}\left(1\right)\oplus\mathfrak{u}\left(1\right)$ | $\mathfrak{so}\left(3\right)\oplus\mathfrak{so}\left(3\right)\oplus\mathfrak{so}\left(3\right)$ | $\mathfrak{so}\left(8\right)$
$\mathbb{\mathfrak{tri}}\left(\mathbb{A}_{s}\right)$ | Ø | $\mathfrak{so}\left(1,1\right)\oplus\mathfrak{so}\left(1,1\right)$ | $\mathfrak{sl}_{2}\left(\mathbb{R}\right)\oplus\mathfrak{sl}_{2}\left(\mathbb{R}\right)\oplus\mathfrak{sl}_{2}\left(\mathbb{R}\right)$ | $\mathfrak{so}\left(4,4\right)$
$\mathfrak{der}\left(\mathbb{A}\right)$ | Ø | Ø | $\mathfrak{so}\left(3\right)$ | $\mathfrak{g}_{2\left(-14\right)}$
$\mathfrak{der}\left(\mathbb{A}_{s}\right)$ | Ø | Ø | $\mathfrak{sl}_{2}\left(\mathbb{R}\right)$ | $\mathfrak{g}_{2\left(2\right)}$
$\mathfrak{A}\left(\mathbb{A}\right)$ | Ø | $\mathfrak{u}_{1}$ | $\mathfrak{su}_{2}$ | Ø
$\mathfrak{A}\left(\mathbb{A}_{s}\right)$ | Ø | $\mathfrak{so}_{1,1}$ | $\mathfrak{sl}_{2}\left(\mathbb{R}\right)$ | Ø
Table 1: Triality and derivation algebras of Hurwitz algebras. Moreover we
also added the algebra
$\mathfrak{A}\left(\mathbb{A}\right):=\mathfrak{tri}\left(\mathbb{A}\right)\ominus\mathfrak{so}\left(\mathbb{A}\right),$
that will be used in Sec. 3.2 in the definition of the Rosenfeld planes.

###  2.3 Euclidean and Lorentzian simple rank-3 Jordan algebras

Let $\mathbb{A}$ be a composition algebra and $H_{n}\left(\mathbb{A}\right)$
be the set of Hermitian $n\times n$ matrices with elements in $\mathbb{A}$
such that

$X^{\dagger}:=\overline{X}^{t}=X,$ (7)

where the conjugation is the one pertaining to $\mathbb{A}$ itself.
When666Excluding $\mathbb{A}=\mathbb{O}$ and $\mathbb{O}_{s}$, one can
actually consider any $n\in\mathbb{N}$. $n=2$ or $3$, we define the simple,
rank-$n$ Euclidean Jordan algebra $\mathfrak{J}_{n}\left(\mathbb{A}\right)$ as
the commutative algebra over $H_{n}\left(\mathbb{A}\right)$ with the Jordan
product $\circ$ defined by the anticommutator

$X\circ
Y:=\frac{1}{2}\left(XY+YX\right)=:\left\\{X,Y\right\\},\leavevmode\nobreak\
\forall X,Y\in H_{n}\left(\mathbb{A}\right),$ (8)

where juxtaposition denotes the standard ’rows-by-columns’ matrix product.
Moreover, we define $\mathfrak{J}_{n}^{\prime}\left(\mathbb{A}\right)$ as the
subspace of the Jordan algebra orthogonal to the identity $I$ (namely, the
subspace of traceless matrices in $\mathfrak{J}_{n}\left(\mathbb{A}\right)$),
endowed with the product inherited777Note that
$\mathfrak{J}_{n}^{\prime}\left(\mathbb{A}\right)$ is not closed under the
restriction of the Jordan product $\circ$ of
$\mathfrak{J}_{n}\left(\mathbb{A}\right)$ to
$\mathfrak{J}_{n}^{\prime}\left(\mathbb{A}\right)$ itself. However, one can
deform $\circ$ into the Michel-Radicati product $\ast$, defined as $X\ast
Y:=X\circ Y-\frac{\text{Tr}\left(X\circ Y\right)}{n}I$ $\forall
X,Y\in\mathfrak{J}_{n}^{\prime}\left(\mathbb{A}\right)$, under which
$\mathfrak{J}_{n}^{\prime}\left(\mathbb{A}\right)$ is closed [MR70, MR73].
from $\mathfrak{J}_{n}\left(\mathbb{A}\right)$ (see e.g. [BS03, p.8]). In
particular, for $n=3$ the simple, rank-$3$ Euclidean Jordan algebra
$\mathfrak{J}_{3}\left(\mathbb{A}\right)$ has elements

$X:=\left(\begin{array}[]{ccc}a&x_{1}&x_{2}\\\ \overline{x}_{1}&b&x_{3}\\\
\overline{x}_{2}&\overline{x}_{3}&c\end{array}\right)\in\mathfrak{J}_{3}\left(\mathbb{A}\right),$
(9)

where $x_{1},x_{2},x_{3}\in\mathbb{A}$ and $a,b,c\in\mathbb{R}$.

The Hermiticity condition (7) can be generalized by inserting a pseudo-
Euclidean metric :

$\displaystyle\eta X^{\dagger}\eta=X,$ (10)
$\displaystyle\eta:=\text{diag}\left(\underset{p\leavevmode\nobreak\
\text{times}}{\underbrace{-1,...,-1}},\underset{n-p\leavevmode\nobreak\
\text{times}}{\underbrace{1,...,1}}\right).$ (11)

Correspondingly, one defines the simple, rank-$n$ pseudo-Euclidean Jordan
algebra
$\mathfrak{J}_{n-p,p}\left(\mathbb{A}\right)\simeq\mathfrak{J}_{p,n-p}\left(\mathbb{A}\right)$.
In particular, by setting $n=3$, the simple, rank-$3$ Lorentzian Jordan
algebra
$\mathfrak{J}_{2,1}\left(\mathbb{A}\right)\simeq\mathfrak{J}_{1,2}\left(\mathbb{A}\right)$,
whose elements read as

$X:=\left(\begin{array}[]{ccc}a&x_{1}&x_{2}\\\ -\overline{x}_{1}&b&x_{3}\\\
-\overline{x}_{2}&\overline{x}_{3}&c\end{array}\right)\in\mathfrak{J}_{1,2}\left(\mathbb{A}\right),$
(12)

where $x_{1},x_{2},x_{3}\in\mathbb{A}$ and $a,b,c\in\mathbb{R}$.

###  2.4 Real forms of the Freudenthal-Tits Magic Square

As anticipated in (1), given $\mathbb{A}_{1}$ and $\mathbb{A}_{2}$ two
composition algebras, the corresponding entry of Tits-Freudenthal Magic Square
$\mathfrak{m}_{3}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)$ is a Lie algebra
that can be realized in at least three different, but equivalent ways, on
which we will now briefly comment (without the explicit definition of the Lie
brackets, that can be found e.g. in [BS03]).

  1. 1.

Tits construction [Ti] is given by the first line of the r.h.s. of (1) :

$\mathfrak{m}_{3}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right):=\mathfrak{der}(\mathbb{A}_{1})\oplus\mathfrak{der}(\mathfrak{J}_{3}(\mathbb{A}_{2}))\,\dot{+}\,\text{Im}\mathbb{A}_{1}\otimes\mathfrak{J}_{3}^{\prime}(\mathbb{A}_{2}),$
(13)

thus involving the derivation Lie algebra of $\mathbb{A}_{1}$ and the
derivation Lie algebra of the simple, rank-$3$ Euclidean Jordan algebra
$\mathfrak{J}_{3}\left(\mathbb{A}_{2}\right)$ over $\mathbb{A}_{2}$. Tits
construction, despite being not manifestly symmetric under the exchange
$\mathbb{A}_{1}\leftrightarrow\mathbb{A}_{2}$, is the most general of the
three constructions presented here, since it holds for any alternative
algebras $\mathbb{A}_{1}$ and $\mathbb{A}_{2}$, as long as it is possible to
define a Jordan algebra over $\mathbb{A}_{2}$ itself. As mentioned above, Tits
construction can be generalized to $\mathfrak{J}_{2,1}(\mathbb{A}_{2})$, thus
yielding888Since
$\mathfrak{J}_{2,1}(\mathbb{A}_{2})\simeq\mathfrak{J}_{1,2}(\mathbb{A}_{2})$,
it holds that
$\mathfrak{m}_{2,1}(\mathbb{A}_{1},\mathbb{A}_{2})\simeq\mathfrak{m}_{1,2}(\mathbb{A}_{1},\mathbb{A}_{2})$.
$\mathfrak{m}_{1,2}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)$.

  2. 2.

A more symmetric approach was pursued by Vinberg [Vin66], who obtained the
formula given by the second line of the r.h.s. of (1),

$\mathfrak{m}_{3}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)=\mathfrak{der}(\mathbb{A}_{1})\oplus\mathfrak{der}(\mathbb{A}_{2})\,\dot{+}\,\mathfrak{sa}(3,\mathbb{A}_{1}\otimes\mathbb{A}_{2}).$
(14)

This formula is manifestly symmetric under the exchange
$\mathbb{A}_{1}\leftrightarrow\mathbb{A}_{2}$, but it only holds if
$\mathbb{A}_{1}$ and $\mathbb{A}_{2}$ are both composition algebras; it
involves only the derivation Lie algebras of $\mathbb{A}$ and $\mathbb{B}$, as
well as the $3\times 3$ antisymmetric matrices over
$\mathbb{A}_{1}\otimes\mathbb{A}_{2}$. As mentioned above, a suitable
deformation of (14) is possible, in order to give rise to
$\mathfrak{m}_{1,2}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)$.

  3. 3.

Another symmetric formula under the exchange
$\mathbb{A}_{1}\leftrightarrow\mathbb{A}_{2}$ was obtained by Barton and
Sudbery [BS03] (also cf. [Ev]), and it is given by the third line of the
r.h.s. of (1),

$\mathfrak{m}_{3}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)=\mathfrak{tri}(\mathbb{A}_{1})\oplus\mathfrak{tri}(\mathbb{A}_{2})\,\dot{+}\,3(\mathbb{A}_{1}\otimes\mathbb{A}_{2}),$
(15)

involving the triality algebras $\mathfrak{tri}\left(\mathbb{A}_{1}\right)$
and $\mathfrak{tri}\left(\mathbb{A}_{2}\right)$, together with three copies of
the tensor product $\mathbb{A}_{1}\otimes\mathbb{A}_{2}$. Again, as mentioned
above, a suitable deformation of (15) is possible, in order to give rise to
$\mathfrak{m}_{1,2}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)$.

Other, different versions of the construction of Freudenthal-Tits Magic Square
have been developed by Santander and Herranz [SH, Sa], Atsuyama [Ats, Atsu2]
and Elduque [El04, El02, El18], all involving composition algebras (even
though Elduque’s construction involves flexible composition algebras instead
of alternative composition algebras [El18]).

###  2.5 Magic Square of order 2

Inspired by the works of Freudenthal and Tits, Barton and Sudbery [BS03] also
considered a different Magic Square, based on $2\times 2$ matrices, and
defined by the following formula :

$\mathfrak{m}_{2}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right):=\mathfrak{so}\left(\mathbb{A}_{1}^{\prime}\right)\oplus\mathfrak{der}\left(\mathfrak{J}_{2}\left(\mathbb{A}_{2}\right)\right)\oplus\left(\mathbb{A}_{1}^{\prime}\otimes\mathfrak{J}_{2}^{\prime}\left(\mathbb{A}_{2}\right)\right),$
(16)

which also enjoys a “Vinberg-like” equivalent version as

$\mathfrak{m}_{2}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)=\mathfrak{so}\left(\mathbb{A}_{1}^{\prime}\right)\oplus\mathfrak{so}\left(\mathbb{A}_{2}^{\prime}\right)\oplus\mathfrak{sa}_{2}\left(\mathbb{A}_{1}\otimes\mathbb{A}_{2}\right),$
(17)

that was more recentely used at Lie group level in [DHK]. Formula (16) can be
generalized to involve simple, rank-2 Lorentzian Jordan algebras
$\mathfrak{J}_{1,1}\left(\mathbb{A}_{2}\right)$ (see Sec. 2.3), thus obtaining
the Lorentzian version of the Magic Square of order 2,

$\mathfrak{m}_{1,1}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right):=\mathfrak{so}\left(\mathbb{A}_{1}^{\prime}\right)\oplus\mathfrak{der}\left(\mathfrak{J}_{1,1}\left(\mathbb{A}_{2}\right)\right)\oplus\left(\mathbb{A}_{1}^{\prime}\otimes\mathfrak{J}_{1,1}^{\prime}\left(\mathbb{A}_{2}\right)\right),$
(18)

which we will explicitly evaluate in the treatment below (for the first time
in literature, to the best of our knowledge).

###  2.6 Octonionic entries of Magic Squares of order 2 and 3

In order to rigorously define all possible octonionic “Rosenfeld planes” and
“Rosenfeld lines” over $\mathbb{R}$ (i.e. all possible real forms thereof), we
need to isolate all octonionic and split-octonionic entries of all real forms
of the Freudenthal-Tits Magic Square (order 3) [CCM, BM, ABDHN, BS03] and of
the Magic Square of order 2 [BS03], i.e. we need to consider the entries
$\mathfrak{m}_{\alpha}\left(\mathbb{A}_{1},\mathbb{A}_{2}\right)$ in which at
least one of $\mathbb{A}_{1}$ and $\mathbb{A}_{2}$ is $\mathbb{O}$ or
$\mathbb{O}_{s}$, for all possible real forms, namely for $\alpha=3$
(Euclidean order 3), $1,2$ (Lorentzian order 3), $2$ (Euclidean order 2) and
$1,1$ (Lorentzian order 2).

The octonionic entries of the Euclidean Magic Squares of order 3
$\mathfrak{m}_{3}$ (i.e., of all Euclidean real forms of Freudenthal-Tits
Magic Square) are

$\mathbb{A}$ | $\mathfrak{m}_{3}\left(\mathbb{A},\mathbb{O}\right)$ | $\mathfrak{m}_{3}\left(\mathbb{A}_{s},\mathbb{O}\right)$ | $\mathfrak{m}_{3}\left(\mathbb{O}_{s},\mathbb{A}\right)$ | $\mathfrak{m}_{3}\left(\mathbb{A}_{s},\mathbb{O}_{s}\right)$
---|---|---|---|---
$\mathbb{R}$ | $\mathfrak{f}_{4\left(-52\right)}$ | $\mathfrak{f}_{4\left(-52\right)}$ | $\mathfrak{f}_{4\left(4\right)}$ | $\mathfrak{f}_{4\left(4\right)}$
$\mathbb{C}$ | $\mathfrak{e}_{6\left(-78\right)}$ | $\mathfrak{e}_{6\left(-26\right)}$ | $\mathfrak{e}_{6\left(2\right)}$ | $\mathfrak{e}_{6\left(6\right)}$
$\mathbb{H}$ | $\mathfrak{e}_{7\left(-133\right)}$ | $\mathfrak{e}_{7\left(-25\right)}$ | $\mathfrak{e}_{7\left(-5\right)}$ | $\mathfrak{e}_{7\left(7\right)}$
$\mathbb{O}$ | $\mathfrak{e}_{8\left(-248\right)}$ | $\mathfrak{e}_{8\left(-24\right)}$ | $\mathfrak{e}_{8\left(-24\right)}$ | $\mathfrak{e}_{8\left(8\right)}$

The entries of the table above comprise all real forms of $\mathfrak{e}_{8}$
and $\mathfrak{e}_{7}$, and most of the real forms of $\mathfrak{e}_{6}$ and
$\mathfrak{f}_{4}$. The missing real forms $\mathfrak{f}_{4\left(-20\right)}$
and $\mathfrak{e}_{6\left(-14\right)}$ can be recovered as the octonionic
entries of the Lorentzian Magic Squares of order 3 $\mathfrak{m}_{2,1}$ (i.e.,
of all Lorentzian real forms of Freudenthal-Tits Magic Square), given by

$\mathbb{A}$ | $\mathcal{\mathfrak{m}}_{1,2}\left(\mathbb{A},\mathbb{O}\right)$ | $\mathcal{\mathfrak{m}}_{1,2}\left(\mathbb{A}_{s},\mathbb{O}\right)$ | $\mathcal{\mathfrak{m}}_{1,2}\left(\mathbb{O}_{s},\mathbb{A}\right)$ | $\mathcal{\mathfrak{m}}_{1,2}\left(\mathbb{A}_{s},\mathbb{O}_{s}\right)$
---|---|---|---|---
$\mathbb{R}$ | $\mathfrak{f}_{4\left(-20\right)}$ | $\mathfrak{f}_{4\left(-20\right)}$ | $\mathfrak{f}_{4\left(4\right)}$ | $\mathfrak{f}_{4\left(4\right)}$
$\mathbb{C}$ | $\mathfrak{e}_{6\left(-14\right)}$ | $\mathfrak{e}_{6\left(-26\right)}$ | $\mathfrak{e}_{6\left(2\right)}$ | $\mathfrak{e}_{6\left(6\right)}$
$\mathbb{H}$ | $\mathfrak{e}_{7\left(-5\right)}$ | $\mathfrak{e}_{7\left(-25\right)}$ | $\mathfrak{e}_{7\left(-5\right)}$ | $\mathfrak{e}_{7\left(7\right)}$
$\mathbb{O}$ | $\mathfrak{e}_{8\left(8\right)}$ | $\mathfrak{e}_{8\left(-24\right)}$ | $\mathfrak{e}_{8\left(-24\right)}$ | $\mathfrak{e}_{8\left(8\right)}$

On the other hand, the octonionic entries of the Euclidean Magic Squares of
order 2 $\mathfrak{m}_{2}$ are999The explicit form of
$\mathfrak{m}_{2}\left(\mathbb{A}_{s},\mathbb{B}_{s}\right)$ is, as far as we
know, not present in the current literature. We will present a detailed
treatment elsewhere, and here we confine ourselves to report its octonionic
column only.

$\mathbb{A}$ | $\mathcal{\mathfrak{m}}_{2}\left(\mathbb{A},\mathbb{O}\right)$ | $\mathcal{\mathfrak{m}}_{2}\left(\mathbb{A}_{s},\mathbb{O}\right)$ | $\mathcal{\mathfrak{m}}_{2}\left(\mathbb{O}_{s},\mathbb{A}\right)$ | $\mathcal{\mathfrak{m}}_{2}\left(\mathbb{A}_{s},\mathbb{O}_{s}\right)$
---|---|---|---|---
$\mathbb{R}$ | $\mathfrak{so}\left(9\right)$ | $\mathfrak{so}\left(9\right)$ | $\mathfrak{so}\left(5,4\right)$ | $\mathfrak{so}\left(5,4\right)$
$\mathbb{C}$ | $\mathfrak{so}\left(10\right)$ | $\mathfrak{so}\left(9,1\right)$ | $\mathfrak{so}\left(6,4\right)$ | $\mathfrak{so}\left(5,5\right)$
$\mathbb{H}$ | $\mathfrak{so}\left(12\right)$ | $\mathfrak{so}\left(10,2\right)$ | $\mathfrak{so}\left(8,4\right)$ | $\mathfrak{so}\left(6,6\right)$
$\mathbb{O}$ | $\mathfrak{so}\left(16\right)$ | $\mathfrak{so}\left(12,4\right)$ | $\mathfrak{so}\left(12,4\right)$ | $\mathfrak{so}\left(8,8\right)$

Finally, the octonionic entries of the Lorentzian Magic Squares of order 2
$\mathfrak{m}_{1,1}$ are given by101010The Lorentzian magic square of order
two yield to three algebras, i.e. $\mathfrak{so}\left(8,1\right)$,
$\mathfrak{so}\left(8,2\right)$ and $\mathfrak{so}\left(8,4\right)$ which are
not covered in other magic squares. The explicit forms of
$\mathfrak{m}_{1,1}\left(\mathbb{A},\mathbb{B}\right)$,
$\mathfrak{m}_{1,1}\left(\mathbb{A}_{s},\mathbb{B}\right)$ and
$\mathfrak{m}_{1,1}\left(\mathbb{A}_{s},\mathbb{B}_{s}\right)$ are, as far as
we know, not present in the current literature. We will present a detailed
treatment elsewhere, and here we confine ourselves to report their octonionic
rows and columns only.

$\mathbb{A}$ | $\mathcal{\mathcal{\mathfrak{m}}}_{1,1}\left(\mathbb{A},\mathbb{O}\right)$ | $\mathcal{\mathcal{\mathfrak{m}}}_{1,1}\left(\mathbb{A}_{s},\mathbb{O}\right)$ | $\mathcal{\mathcal{\mathfrak{m}}}_{1,1}\left(\mathbb{O}_{s},\mathbb{A}\right)$ | $\mathcal{\mathcal{\mathfrak{m}}}_{1,1}\left(\mathbb{A}_{s},\mathbb{O}_{s}\right)$
---|---|---|---|---
$\mathbb{R}$ | $\mathfrak{so}\left(8,1\right)$ | $\mathfrak{so}\left(8,1\right)$ | $\mathfrak{so}\left(5,4\right)$ | $\mathfrak{so}\left(5,4\right)$
$\mathbb{C}$ | $\mathfrak{so}\left(8,2\right)$ | $\mathfrak{so}\left(9,1\right)$ | $\mathfrak{so}\left(6,4\right)$ | $\mathfrak{so}\left(5,5\right)$
$\mathbb{H}$ | $\mathfrak{so}\left(8,4\right)$ | $\mathfrak{so}\left(10,2\right)$ | $\mathfrak{so}\left(8,4\right)$ | $\mathfrak{so}\left(6,6\right)$
$\mathbb{O}$ | $\mathfrak{so}\left(8,8\right)$ | $\mathfrak{so}\left(12,4\right)$ | $\mathfrak{so}\left(12,4\right)$ | $\mathfrak{so}\left(8,8\right)$

##  3 “Magic” formulæ for Rosenfeld planes

In his study of the geometry of Lie groups [Ro97], Rosenfeld defined its
“projective” planes $\left(\mathbb{A}\otimes\mathbb{B}\right)P^{2}$ as the
completion of some affine planes obtained as non-associative modules over the
tensor algebra $\mathbb{A}\otimes\mathbb{B}$. He then argued the form of the
matrices composing the linear transformations associated with the Lie algebra
of the collineations that preserved the polarity, i.e. the isometries of the
plane. This approach allowed him to relate all real forms of exceptional Lie
groups [Ro93, Ro98] with isometries of suitable “projective” hyperbolic
spaces. As mentioned above, unfortunately the procedure carried out by
Rosenfeld was not rigorous, since many of theorems used in [Ro97] do not
extend to the case of algebras that are not alternative nor of composition, as
it is in the case of many algebras listed in Table 2.

We present here a general and rigorous way to define the spaces considered by
Rosenfeld, in terms of their isometry and isotropy groups, namely using the
theory of coset manifolds.

###  3.1 Coset manifolds

Since our “magic” formulæ define all Rosenfeld planes as coset manifolds, it
is worth briefly reviewing them, and their relation to homogenous spaces. An
_homogeneous space_ is a manifold on which a Lie group acts transitively, i.e.
a manifold on which is defined an action $\rho_{g}$ from $G\times M$ in $M$
such that $\rho_{e}\left(m\right)=m$ for $e$ the identity in $G$ and $m\in M$
and for which, given any $m,n\in M$ it exists a a not necessarely unique $g\in
G$ such that $\rho_{g}\left(m\right)=n$. Within this framework, the _isotropy
group_ $\text{Isot}_{m}\left(G\right)$ is the set formed by the elements of
$G$ that fix the point $m\in M$ under the action of $G$, i.e.

$\text{Isot}_{m}\left(G\right)=\left\\{g\in
G:\rho_{g}\left(m\right)=m\right\\}.$ (19)

Since, by definition of group action, we have that $\rho_{e}\left(m\right)=m$
and $\rho_{gh}\left(m\right)=\rho_{g}\left(\rho_{h}\left(m\right)\right)$,
then $K=\text{Isot}_{m}\left(G\right)$ is a closed subgroup of $G$ and,
moreover the natural map from the quotient space $G/K$ in $M$ given by
$gK\mapsto gm$ is a diffeomorphism (see [Ar] and Refs. therein).

Algebra | Comm. | Ass. | Alter. | Flex. | Pow. Ass.
---|---|---|---|---|---
$\mathbb{C}\otimes\mathbb{C}$ | Yes | Yes | Yes | Yes | Yes
$\mathbb{C}\otimes\mathbb{H}$ | No | Yes | Yes | Yes | Yes
$\mathbb{H}\otimes\mathbb{H}$ | No | Yes | Yes | Yes | Yes
$\mathbb{C}\otimes\mathbb{O}$ | No | No | Yes | Yes | Yes
$\mathbb{H}\otimes\mathbb{O}$ | No | No | No | No | No
$\mathbb{O}\otimes\mathbb{O}$ | No | No | No | No | No
Table 2: Properties of the algebra $\mathbb{A}\otimes\mathbb{B}$ where
$\mathbb{A},\mathbb{B}$ are Hurwitz algebras. As for the property an algebra
$\mathbb{A}$ is said to be _commutative_ if $xy=yx$ for every $x,y\in X$; it
is defined as _associative_ if satisfies $x\left(yz\right)=\left(xy\right)z$;
_alternative_ if $x\left(yx\right)=\left(xy\right)x$; _flexible_ if
$x\left(yy\right)=\left(xy\right)y$ and, finally, _power-associative_ if
$x\left(xx\right)=\left(xx\right)x$.

Given a Lie group $G$ and a closed subgroup $K<G$, then the coset space
$G/K=\left\\{gK:g\in G\right\\}$ is endowed with a natural manifold structure
inherited by $G$ and is, therefore, called a _coset manifold_. Notice that the
action of $G$ on the coset manifold $G/K$, given by the translation $\tau_{g}$
defined as

$\tau_{g}\left(m\right)=gm,$ (20)

for every $g\in G$ and $m\in G/K$, is _transitive_ , i.e. for every $m,n\in
G/K$ it exists a (not necessarily unique) $g\in G$ such that $n=gm$, and thus
the coset manifold $G/K$ is an homogenous space. On the other hand, since
multiple groups can act transitively on the same manifold with different
isotropy groups, then an homogeneous space can be realised in multiple way as
a coset manifold. Moreover, a close look to the definitions shows that the
isotropy group $\text{Isot}_{m}\left(G/K\right)$ is exactly the closed
subgroup $K$.111111In particular, the origin of $G/K$ is, by definition, the
point at which the $K$-invariance is immediately manifest. In general, the
holonomy subgroup and the isotropy subgroup have the same identity-connected
component; so, if one assumes that $G/K$ is simply-connected, they are equal
(see e.g. [Be, He] and Refs. therein).

Moreover, let $G$ be a Lie group and $K$ a closed and connected subgroup of
$G$, denoting with $\mathfrak{g}$ and $\mathfrak{k}$ their respective Lie
algebras, then the coset manifold $G/K$ is _reductive_ if there exists a
subspace $\mathfrak{m}$ such that
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}$ and

$\displaystyle\begin{cases}\left[\mathfrak{k},\mathfrak{k}\right]&\subset\mathfrak{k},\\\
\left[\mathfrak{k},\mathfrak{m}\right]&\subset\mathfrak{m},\end{cases}$ (21)

while if in addition to (21) we also have

$\left[\mathfrak{m},\mathfrak{m}\right]\subset\mathfrak{k},$ (22)

then the space is _symmetric_. All Rosenfeld planes are symmetric coset
manifolds.

It is also worth noting that for any coset manifold $G/K$ the structure
constants of the Lie algebra $\mathfrak{g}$ of the Lie group $G$ define
completely the structure constants of the manifold, thus the invariant
metrics, and all the metric-dependent tensors, such as the curvature tensor,
the Ricci tensor, etc. Indeed, let $\left\\{E_{1},...,E_{n}\right\\}$ be a
basis for $\mathfrak{g}$ in such a way that $\left\\{E_{1},...,E_{m}\right\\}$
are a basis for $\mathfrak{k}$, which we will also call
$\left\\{K_{1},...,K_{m}\right\\}$ for readibility reasons, and
$\left\\{E_{m+1},...,E_{n}\right\\}$ a basis for $\mathfrak{m}$ that we will
also denote as $\left\\{M_{m+1},...,M_{n}\right\\}$. Then consider the
structure constants of the algebra $\mathfrak{g}$, i.e.

$\left[E_{j},E_{k}\right]=\stackrel{{\scriptstyle[}}{{i}}=1]{n}{\sum}C_{jk}^{i}E_{i},$
(23)

for every $j,k\in\left\\{1,...,n\right\\}$. Conditions for reductivity in (21)
are then translated in

$\begin{cases}\left[K_{j},K_{k}\right]=\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}C_{jk}^{i}K_{i}&\text{for
}j,k\in\left\\{1,...,m\right\\},\\\
\left[K_{j},M_{k}\right]=\stackrel{{\scriptstyle[}}{{i}}=m+1]{n}{\sum}C_{jk}^{i}M_{i}&\text{for
}j\in\left\\{1,...,m\right\\},k\in\left\\{m+1,...,n\right\\},\end{cases}$ (24)

while, on the other hand, from

$\left[M_{j},M_{k}\right]=\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}C_{jk}^{i}K_{i}+\stackrel{{\scriptstyle[}}{{i}}=m+1]{n}{\sum}C_{jk}^{i}M_{i},$
(25)

we deduce that if $C_{jk}^{i}=0$ for all $i,j,k\in\left\\{m+1,...,n\right\\}$,
we have also a symmetric space. From the structure constants $C_{jk}^{i}$, all
geometrical invariants of the coset manifold can be obtained (see [FF17] for
all technical details) such as the Riemann tensor over the coset manifold
$G/K$ which is given by

$R_{bcd}^{a}=\stackrel{{\scriptstyle[}}{{e}}=m+1]{n}{\sum}\frac{1}{\lambda^{2}}\left(\frac{1}{8}C_{ed}^{a}C_{bc}^{e}-\frac{1}{8\lambda}C_{be}^{a}C_{cd}^{e}-C_{ec}^{a}C_{bd}^{e}\right)-\frac{1}{2\lambda^{2}}\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}C_{bi}^{a}C_{cd}^{i},$
(26)

and that, in case of symmetric spaces, i.e. when $C_{jk}^{i}=0$ for
$i,j,k\in\left\\{m+1,...,n\right\\}$, reduces drastically to

$R_{bcd}^{a}=-\frac{1}{2\lambda^{2}}\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}C_{bi}^{a}C_{cd}^{i},$
(27)

for every $a,b,c,d\in\left\\{m+1,...,n\right\\}$.

In fact, what is relevant for our purposes is that the Lie group $G$, which
acts as isometry group, and its closed subgroup $K$, which acts as isotropy
group, define completely all the metric properties and geometric invariants of
the Rosenfeld planes. On the other hand, instead of working on Lie groups $G$
and $K$, i.e. with the isometry group and the isotropy group respectively, in
the following sections we will work with their respective Lie algebras
$\mathfrak{g}$ and $\mathfrak{k}$, in order to recover the appropriate $G$ and
$K$ and thus define the Rosenfeld plane.

###  3.2 “Magic” formulæ

As seen in the previous section, the geometry of a coset manifold is fully
determined by the (global) isotropy Lie group $G$ and by its (local) subgroup
given by the isotropy Lie group $K$. We now characterize the Rosenfeld
“projective” planes by specifying the real form of the Lie groups that will
act as isometry and isotropy groups, while the Lie brackets - and therefore
the metrical properties of the space - are those arising from Tits
construction. It is here worth noting that the classification of the real
forms of simple Lie groups makes use of the character $\chi$, defined as the
difference of the cardinality of non-compact and compact generators, i.e.
$\chi:=\\#_{nc}-\\#_{c}$; consequently, Rosenfeld planes will have a
corresponding $\chi$.

Let $\mathbb{A}$ and $\mathbb{B}$ be two Hurwitz algebras, in their division
or split versions. Moreover, let the Lie group
$\mathcal{A}\left(\mathbb{A}\right)$ be such that its Lie algebra is
Lie$\left(\mathcal{A}\left(\mathbb{A}\right)\right)\equiv\mathfrak{A}(\mathbb{A}):=\mathfrak{tri}\left(\mathbb{A}\right)\ominus\mathfrak{so}\left(\mathbb{A}\right)$,
cfr. Table 1. Moreover, let
$\mathcal{M}_{\alpha}\left(\mathbb{A},\mathbb{B}\right)$ be the Lie group with
Lie algebra given by the $\left(\mathbb{A},\mathbb{B}\right)$-entry of the
Magic Square $\mathfrak{m}_{\alpha}\left(\mathbb{A},\mathbb{B}\right)$,
namely121212For the pseudo-orthogonal Lie algebras, we will generally consider
the spin covering of the corresponding Lie group.
Lie$\left(\mathcal{M}_{\alpha}\left(\mathbb{A},\mathbb{B}\right)\right)=\mathfrak{m}_{\alpha}\left(\mathbb{A},\mathbb{B}\right)$.
In order to characterize “projective” Rosenfeld planes over tensor products of
Hurwitz algebras in terms of coset manifolds with isometry (resp. isotropy)
Lie groups whose Lie algebras are entries of real forms of the Magic Square of
order 3 (resp. 2), we now introduce the following three different classes of
locally symmetric, (pseudo-)Riemannian coset manifolds, that we name as
Rosenfeld planes :

  1. 1.

The projective Rosenfeld plane

$\left(\mathbb{A}\otimes\mathbb{B}\right)P^{2}\simeq\frac{\mathcal{M}_{3}\left(\mathbb{A},\mathbb{B}\right)}{\mathcal{M}_{2}\left(\mathbb{A},\mathbb{B}\right)\otimes\mathcal{A}\left(\mathbb{A}\right)\otimes\mathcal{A}\left(\mathbb{B}\right)}.$
(28)

  2. 2.

The hyperbolic Rosenfeld plane

$\left(\mathbb{A}\otimes\mathbb{B}\right)H^{2}\simeq\frac{\mathcal{M}_{1,2}\left(\mathbb{A},\mathbb{B}\right)}{\mathcal{M}_{2}\left(\mathbb{A},\mathbb{B}\right)\otimes\mathcal{A}\left(\mathbb{A}\right)\otimes\mathcal{A}\left(\mathbb{B}\right)}.$
(29)

  3. 3.

The pseudo-Rosenfeld plane

$\left(\mathbb{A}\otimes\mathbb{B}\right)\widetilde{H}^{2}\simeq\frac{\mathcal{M}_{1,2}\left(\mathbb{A},\mathbb{B}\right)}{\mathcal{M}_{1,1}\left(\mathbb{A},\mathbb{B}\right)\otimes\mathcal{A}\left(\mathbb{A}\right)\otimes\mathcal{A}\left(\mathbb{B}\right)}.$
(30)

Eqs. (28), (29) and (30) are named “magic” formulæ, since they characterize
the Rosenfeld planes as homogeneous (symmetric) manifolds, with isometry
(resp. isotropy) groups whose Lie algebras are given by the entries of some
real forms of the Magic Square of order 3 (resp. 2), with further isotropy
factors given by the Lie groups $\mathcal{A}$ associated to the algebras
$\mathbb{A}$ and $\mathbb{B}$ defining the tensor product associated to the
class of Rosenfeld plane under consideration. As previously noticed, the term
“projective” and “hyperbolic” are here to be intended in a vague sense since
none of the octonionic Rosenfeld planes with dimension greater than 16 satisfy
axioms of projective or hyperbolic geometry. Nevertheless, those adjectives
are not arbitrary since the notion of such projective planes can be made
precise as in [Atsu2, AB] and it then agrees with the above definitions.

##  4 Octonionic Rosenfeld planes

We now consider the “magic” formulæ (28)-(30) in the cases131313As mentioned
above, this restriction does not imply any loss of generality, as far as the
other, non-octonionic Rosenfeld planes can be obtained as suitable sub-
manifolds of the octonionic Rosenfeld planes. in which $\mathbb{A}$ and/or
$\mathbb{B}$ is $\mathbb{O}$ or $\mathbb{O}_{s}$ : this will allow us to
rigorously introduce the octonionic Rosenfeld planes, which all share the fact
that their isometry group is a real form of an exceptional Lie group of F- or
E- type; however, it is here worth anticipating that a few real forms of
Rosenfeld planes with exceptional isometry groups cannot be characterized in
this way.

###  4.1 $\mathbb{O}\simeq\mathbb{R}\otimes\mathbb{O}$

The simplest case concerns the tensor product $\mathbb{R}\otimes\mathbb{O}$,
which is nothing but $\mathbb{O}$ : in fact, this was the original observation
by Rosenfeld that started it all, yielding to the usual octonionic projective,
split-octonionic and hyperbolic plane, i.e. $\mathbb{O}P^{2}$,
$\mathbb{O}H^{2}$ and $\mathbb{O}_{s}P^{2}$, respectively. Within the
framework introduced above, the starting point is given by the Cayley-Moufang
plane over $\mathbb{C}$, namely by the octonionic projective plane over
$\mathbb{C}$, i.e. by the locally symmetric coset manifold having as isometry
group the complex form of $\text{F}_{4}$, and as isotropy group
$\text{Spin}\left(9,\mathbb{C}\right)$, i.e.

$\mathbb{O}P_{\mathbb{C}}^{2}\simeq\frac{\text{F}_{4}^{\mathbb{C}}}{\text{Spin}\left(9,\mathbb{C}\right)}.$
(31)

In this coset space formulation, the tangent space of
$\mathbb{O}P_{\mathbb{C}}^{2}$ can be identified with the
$\boldsymbol{16}_{\mathbb{C}}$-dimensional spinor representation space of the
isotropy group $\text{Spin}\left(9,\mathbb{C}\right)$.

Then, by specifying the formulæ (28), (29) and (30) for
$\mathbb{A}=\mathbb{R}$ and $\mathbb{B}=\mathbb{O}$ or $\mathbb{O}_{s}$, we
obtain all real forms141414By real forms of $\mathbb{O}P_{\mathbb{C}}^{2}$, we
here mean the cosets with isometry groups given by all real (compact and non-
compact) forms of $F_{4}$, and with isotropy group given by all (compact and
non-compact) real forms of Spin$(9)$ which are subgroups of the corresponding
real form of $F_{4}$. of (31) as Rosenfeld planes over
$\mathbb{R}\otimes\mathbb{O}\simeq\mathbb{O}$ or over
$\mathbb{R}\otimes\mathbb{O}_{s}\simeq\mathbb{O}_{s}$; they are summarized by
the following151515Recall that
$\mathfrak{A}(\mathbb{R})=\mathfrak{A}(\mathbb{O})=\mathfrak{A}(\mathbb{O}_{s})=\varnothing$.
Table [CMCAb] :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\mathbb{O}P^{2}$ | $\text{F}_{4(-52)}$ | $\text{Spin}\left(9\right)$ | $0$ | $16$ | $-16$
$\mathbb{O}H^{2}$ | $\text{F}_{4(-20)}$ | $\text{Spin}\left(9\right)$ | $16$ | $0$ | $16$
$\mathbb{O}\widetilde{H}^{2}$ | $\text{F}_{4(-20)}$ | $\text{Spin}\left(8,1\right)$ | $8$ | $8$ | $0$
$\mathbb{O}_{s}P^{2}$ | $\text{F}_{4(4)}$ | $\text{Spin}\left(5,4\right)$ | $8$ | $8$ | $0$

along with

$\mathbb{O}_{s}P^{2}\simeq\mathbb{O}_{s}H^{2}\simeq\mathbb{O}_{s}\widetilde{H}^{2}.$
(32)

###  4.2 $\mathbb{C}\otimes\mathbb{O}$

In the bioctonionic case, i.e. for $\mathbb{C}\otimes\mathbb{O}$, the starting
point is given by the bioctonionic projective plane over $\mathbb{C}$, i.e. by
the locally symmetric coset manifold having as isometry group the complex form
of $E_{6}$, and as isotropy group Spin$\left(10,\mathbb{C}\right)\otimes$U1,
i.e.

$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}\simeq\frac{\text{E}_{6}^{\mathbb{C}}}{\text{Spin}\left(10,\mathbb{C}\right)\otimes\left(\text{U}_{1}\right)_{\mathbb{C}}},$
(33)

which is a Kähler manifold, and has been recently treated in [CMCAa]. In this
case, the tangent space of the coset manifold (33) is the
$\left(\mathbf{16}_{\mathbb{C},+}\oplus\overline{\mathbf{16}}_{\mathbb{C},-}\right)$
representation of the isotropy group
$\text{Spin}\left(10,\mathbb{C}\right)\otimes\text{U}_{1}$.

Then, by specifying the formulæ (28), (29) and (30) for
$\mathbb{A}=\mathbb{C}$ or $\mathbb{C}_{s}$ and $\mathbb{B}=\mathbb{O}$ or
$\mathbb{O}_{s}$, we obtain all real forms161616By real forms of
$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}$, we here mean the
cosets with isometry groups given by all real (compact and non-compact) forms
of $E_{6}$, and with isotropy group given by all (compact and non-compact)
real forms of
Spin$\left(10,\mathbb{C}\right)\otimes\left(\text{U}_{1}\right)_{\mathbb{C}}$
which are subgroups of the corresponding real form of $E_{6}$. of (33) which
can be expressed as Rosenfeld planes over $\mathbb{C}$(or$\leavevmode\nobreak\
\mathbb{C}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$);
they are summarized by the following171717Recall that
$\mathfrak{A}(\mathbb{C})=\mathfrak{u}_{1}$, and
$\mathfrak{A}(\mathbb{C}_{s})=\mathfrak{so}_{1,1}$. Table :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\mathbb{\left(\mathbb{C\otimes O}\right)}P^{2}$ | $\text{E}_{6(-78)}$ | $\text{Spin}\left(10\right)\otimes\text{U}_{1}$ | $0$ | $32$ | $-32$
$\mathbb{\left(\mathbb{C\otimes O}\right)}H^{2}$ | $\text{E}_{6(-14)}$ | $\text{Spin}\left(10\right)\otimes\text{U}_{1}$ | $32$ | $0$ | $32$
$\mathbb{\left(\mathbb{C\otimes O}\right)}\widetilde{H}^{2}$ | $\text{E}_{6(-14)}$ | $\text{Spin}\left(8,2\right)\otimes\text{U}_{1}$ | $16$ | $16$ | $0$
$\left(\mathbb{C\otimes O}_{s}\right)P^{2}$ | $\text{E}_{6(2)}$ | $\text{Spin}\left(6,4\right)\otimes\text{U}_{1}$ | $16$ | $16$ | $0$
$\left(\mathbb{C}_{s}\mathbb{\otimes O}\right)P^{2}$ | $\text{E}_{6(-26)}$ | $\text{Spin}\left(9,1\right)\otimes\text{SO}\left(1,1\right)$ | $16$ | $16$ | $0$
$\left(\mathbb{C}_{s}\mathbb{\otimes O}_{s}\right)P^{2}$ | $\text{E}_{6(6)}$ | $\text{Spin}\left(5,5\right)\otimes\text{SO}\left(1,1\right)$ | $16$ | $16$ | $0$

along with

$\displaystyle\left(\mathbb{C\otimes O}_{s}\right)P^{2}$
$\displaystyle\simeq\left(\mathbb{C\otimes
O}_{s}\right)H^{2}\simeq\left(\mathbb{C\otimes
O}_{s}\right)\widetilde{H}^{2},$ (34)
$\displaystyle\left(\mathbb{C}_{s}\mathbb{\otimes O}\right)P^{2}$
$\displaystyle\simeq\left(\mathbb{C}_{s}\mathbb{\otimes
O}\right)H^{2}\simeq\left(\mathbb{C}_{s}\mathbb{\otimes
O}\right)\widetilde{H}^{2},$ (35)
$\displaystyle\left(\mathbb{C}_{s}\mathbb{\otimes O}_{s}\right)P^{2}$
$\displaystyle\simeq\left(\mathbb{C}_{s}\mathbb{\otimes
O}_{s}\right)H^{2}\simeq\left(\mathbb{C}_{s}\mathbb{\otimes
O}_{s}\right)\widetilde{H}^{2},$ (36)

expressing the fact that projective, hyperbolic and pseudo Rosenfeld planes
involving $\mathbb{C}_{s}$ and/or $\mathbb{O}_{s}$ are all isomorphic.

The first four manifolds of the above Table, having a U1 factor in the
isotropy group, are Kähler manifolds, whereas the last two, having a SO$(1,1)$
factor in the isotropy group, are pseudo-Kähler manifolds.

Finally, and more importantly, there are two real forms of (33), namely the
locally symmetric, pseudo-Riemannian coset Kähler manifolds

$\displaystyle X_{32,I}$
$\displaystyle:=\frac{\text{E}_{6(2)}}{\text{SO}^{\ast}(10)\otimes\text{U}_{1}},\leavevmode\nobreak\
\\#_{nc}=20,\\#_{c}=12\Rightarrow\chi=8;$ (37) $\displaystyle X_{32,II}$
$\displaystyle:=\frac{\text{E}_{6(-14)}}{\text{SO}^{\ast}(10)\otimes\text{U}_{1}},\leavevmode\nobreak\
\\#_{nc}=12,\\#_{c}=20\Rightarrow\chi=-8,$ (38)

whose isotropy Lie group has the corresponding Lie algebra which is not an
entry of any real form of the Magic Square of order 2.

In other words, since the Lie algebra $\mathfrak{so}^{\ast}(10)$ does not
occur in any real form of the Magic Square of order 2 (see Secs. 2.5 and 2.6),
the symmetric Kähler manifolds (37) and (38) cannot be characterized as
Rosenfeld planes over $\mathbb{C}$(or$\leavevmode\nobreak\
\mathbb{C}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$).

###  4.3 $\mathbb{H}\otimes\mathbb{O}$

In the quaternoctonionic case, i.e. for $\mathbb{H}\otimes\mathbb{O}$, the
starting point is given by the quaternoctonionic “projective” plane over
$\mathbb{C}$, i.e. by the locally symmetric coset manifold having as isometry
group the complex form of $E_{7}$, and as isotropy group
Spin$\left(12,\mathbb{C}\right)\otimes\text{SL}\left(2,\mathbb{C}\right)$,
i.e.

$\left(\mathbb{H}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}\simeq\frac{\text{E}_{7}^{\mathbb{C}}}{\text{Spin}\left(12,\mathbb{C}\right)\otimes\text{SL}\left(2,\mathbb{C}\right)},$
(39)

which is a quaternionic Kähler manifold, and whose tangent space is given by
the $\left(\mathbf{32}^{(\prime)},\mathbf{2}\right)_{\mathbb{C}}$
representation181818For the possible priming of the semispinor $\mathbf{32}$
of Spin$(12)$, see e.g. [Min]. of the isotropy group
$\text{Spin}\left(12,\mathbb{C}\right)\otimes\text{SL}\left(2,\mathbb{C}\right)$.

Then, by specifying the formulæ (28), (29) and (30) for
$\mathbb{A}=\mathbb{H}$ or $\mathbb{H}_{s}$ and $\mathbb{B}=\mathbb{O}$ or
$\mathbb{O}_{s}$, we obtain all real forms191919By real forms of
$\left(\mathbb{H}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}$, we here mean the
cosets with isometry groups given by all real (compact and non-compact) forms
of $E_{7}$, and with isotropy group given by all (compact and non-compact)
real forms of Spin$\left(12,\mathbb{C}\right)\otimes
SL_{2}\left(\mathbb{C}\right)$ which are subgroups of the corresponding real
form of $E_{7}$. of (39) which can be expressed as Rosenfeld planes over
$\mathbb{H}$(or$\leavevmode\nobreak\
\mathbb{H}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$);
they are summarized by the following202020Recall that
$\mathfrak{A}(\mathbb{H})=\mathfrak{su}_{2}$, and
$\mathfrak{A}(\mathbb{H}_{s})=\mathfrak{sl}_{2}(\mathbb{R})$. Table :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\mathbb{\left(\mathbb{H\otimes O}\right)}P^{2}$ | $\text{E}_{7(-133)}$ | $\text{Spin}\left(12\right)\otimes\text{SU}\left(2\right)$ | $0$ | $64$ | $-64$
$\mathbb{\left(\mathbb{H\otimes O}\right)}H^{2}$ | $\text{E}_{7(-5)}$ | $\text{Spin}\left(12\right)\otimes\text{SU}\left(2\right)$ | $64$ | $0$ | $64$
$\mathbb{\left(\mathbb{H\otimes O}\right)}\widetilde{H}^{2}$ | $\text{E}_{7(-5)}$ | $\text{Spin}\left(8,4\right)\otimes\text{SU}\left(2\right)$ | $32$ | $32$ | $0$
$\left(\mathbb{H}_{s}\mathbb{\otimes O}\right)P^{2}$ | $\text{E}_{7(-25)}$ | $\text{Spin}\left(10,2\right)\otimes\text{SL}\left(2,\mathbb{R}\right)$ | $32$ | $32$ | $0$
$\left(\mathbb{H}_{s}\mathbb{\otimes O}_{s}\right)P^{2}$ | $\text{E}_{7(7)}$ | $\text{Spin}\left(6,6\right)\otimes\text{SL}\left(2,\mathbb{R}\right)$ | $32$ | $32$ | $0$

along with

$\displaystyle\mathbb{\left(\mathbb{H\otimes O}\right)}\widetilde{H}^{2}$
$\displaystyle\simeq\left(\mathbb{H\otimes
O}_{s}\right)P^{2}\simeq\left(\mathbb{H\otimes
O}_{s}\right)H^{2}\simeq\left(\mathbb{H}\mathbb{\otimes
O}_{s}\right)\widetilde{H}^{2},$ (40)
$\displaystyle\left(\mathbb{H}_{s}\mathbb{\otimes O}\right)P^{2}$
$\displaystyle\cong\left(\mathbb{H}_{s}\mathbb{\otimes
O}\right)H^{2}\cong\left(\mathbb{H}_{s}\mathbb{\otimes
O}\right)\widetilde{H}^{2},$ (41)
$\displaystyle\left(\mathbb{H}_{s}\mathbb{\otimes O}_{s}\right)P^{2}$
$\displaystyle\cong\left(\mathbb{H}_{s}\mathbb{\otimes
O}_{s}\right)H^{2}\cong\left(\mathbb{H}_{s}\mathbb{\otimes
O}_{s}\right)\widetilde{H}^{2}.$ (42)

The planes that do not involve split algebras, namely the ones having a
SU$(2)$ factor in the isotropy group, are quaternionic Kähler manifolds,
whereas all the ones involving split algebras, namely the ones having a
SL$(2,\mathbb{R})$ factor in the istropy group, are para-quaternionic Kähler
manifolds.

Again, there are three real forms of (39), namely the locally symmetric,
pseudo-Riemannian (para-)quaternionic coset manifolds

$\displaystyle X_{64,I}$
$\displaystyle:=\frac{\text{E}_{7(7)}}{\text{SO}^{\ast}\left(12\right)\otimes\text{SU}\left(2\right)},\leavevmode\nobreak\
\\#_{nc}=40,\\#_{c}=24\Rightarrow\chi=16;$ (43) $\displaystyle X_{64,II}$
$\displaystyle:=\frac{\text{E}_{7(-5)}}{\text{SO}^{\ast}\left(12\right)\otimes\text{SL}\left(2,\text{$\mathbb{R}$}\right)},\leavevmode\nobreak\
\\#_{nc}=32,\\#_{c}=32\Rightarrow\chi=0;$ (44) $\displaystyle X_{64,III}$
$\displaystyle:=\frac{\text{E}_{7(-25)}}{\text{SO}^{\ast}\left(12\right)\otimes\text{SU}\left(2\right)},\leavevmode\nobreak\
\\#_{nc}=24,\\#_{c}=40\Rightarrow\chi=-16,$ (45)

whose isotropy Lie group -up to the factor SU$\left(2\right)$ or
SL$\left(2,\text{$\mathbb{R}$}\right)$\- has the corresponding Lie algebra
which is not an entry of any real form of the Magic Square of order 2.

In other words, since the Lie algebra $\mathfrak{so}^{\ast}(12)$ does not
occur in any real form of the Magic Square of order 2 (see Secs. 2.5 and 2.6),
the symmetric (para-)quaternionic Kähler manifolds (43)-(45) cannot be
characterized as Rosenfeld planes over $\mathbb{H}$(or$\leavevmode\nobreak\
\mathbb{H}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$).
It should however be noticed that the isotropy group of (43)-(45) admits an
interpretation in terms of real forms of the Magic Square of order 3, namely

$\displaystyle X_{64,I}\simeq$
$\displaystyle\frac{\mathcal{M}_{3}\left(\mathbb{H}_{s},\mathbb{O}_{s}\right)}{\mathcal{M}_{3}\left(\mathbb{H}_{s},\mathbb{H}\right)\otimes\mathcal{A}(\mathbb{H})},$
(46) $\displaystyle X_{64,II}\simeq$
$\displaystyle\frac{\mathcal{M}_{3}\left(\mathbb{H},\mathbb{O}_{s}\right)}{\mathcal{M}_{3}\left(\mathbb{H},\mathbb{H}_{s}\right)\otimes\mathcal{A}(\mathbb{H}_{s})},$
(47) $\displaystyle X_{64,III}\simeq$
$\displaystyle\frac{\mathcal{M}_{3}\left(\mathbb{H}_{s},\mathbb{O}\right)}{\mathcal{M}_{3}\left(\mathbb{H}_{s},\mathbb{H}\right)\otimes\mathcal{A}(\mathbb{H})},$
(48)

but still the rationale (if any) of such formulæ is missing, and it is surely
not the one underlying the “magic” formulæ (28)-(30).

###  4.4 $\mathbb{O}\otimes\mathbb{O}$

In the octooctonionic case, i.e. for $\mathbb{O}\otimes\mathbb{O}$, the
starting point is given by the octoooctonionic “projective” plane over
$\mathbb{C}$, i.e. by the locally symmetric coset manifold having as isometry
group the complex form of $\text{E}_{8}$, and as isotropy group
Spin$\left(16,\mathbb{C}\right)$, i.e.

$\left(\mathbb{O}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}\simeq\frac{\text{E}_{8}^{\mathbb{C}}}{\text{Spin}\left(16,\mathbb{C}\right)},$
(49)

whose tangent space is given by the
$\left(\boldsymbol{128}^{(\prime)}\right)_{\mathbb{C}}$
representation212121Again, for the possible priming of the semispinor
$\mathbf{128}$ of Spin$(16)$, see e.g. [Min]. of
$\text{Spin}\left(16,\mathbb{C}\right)$.

Then, by specifying the formulæ (28), (29) and (30) for
$\mathbb{A}=\mathbb{O}$ or $\mathbb{O}_{s}$ and $\mathbb{B}=\mathbb{O}$ or
$\mathbb{O}_{s}$, we obtain all real forms222222By real forms of
$\left(\mathbb{O}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}$, we here mean the
cosets with isometry groups given by all real (compact and non-compact) forms
of $E_{8}$, and with isotropy group given by all (compact and non-compact)
real forms of Spin$\left(16,\mathbb{C}\right)$ which are subgroups of the
corresponding real form of $E_{8}$. of (49) which can be expressed as
Rosenfeld planes over $\mathbb{O}$(or$\leavevmode\nobreak\
\mathbb{O}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$);
they are summarized by the following Table :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\mathbb{\left(\mathbb{O\otimes O}\right)}P^{2}$ | $\text{E}_{8(-248)}$ | $\text{Spin}\left(16\right)$ | $0$ | $128$ | $-128$
$\mathbb{\left(\mathbb{O\otimes O}\right)}H^{2}$ | $\text{E}_{8(8)}$ | $\text{Spin}\left(16\right)$ | $128$ | $0$ | $128$
$\mathbb{\left(\mathbb{O\otimes O}\right)}\widetilde{H}^{2}$ | $\text{E}_{8(8)}$ | $\text{Spin}\left(8,8\right)$ | $64$ | $64$ | $0$
$\left(\mathbb{O}_{s}\mathbb{\otimes O}\right)P^{2}$ | $\text{E}_{8(-24)}$ | $\text{Spin}\left(12,4\right)$ | $64$ | $64$ | $0$

along with

$\displaystyle\mathbb{\left(\mathbb{O\otimes O}\right)}\widetilde{H}^{2}$
$\displaystyle\simeq\left(\mathbb{O}_{s}\mathbb{\otimes
O}_{s}\right)P^{2}\simeq\left(\mathbb{O}_{s}\mathbb{\otimes
O}_{s}\right)H^{2}\simeq\left(\mathbb{O}_{s}\mathbb{\otimes
O}_{s}\right)\widetilde{H}^{2},$ (50) $\displaystyle\left(\mathbb{O\otimes
O}_{s}\right)P^{2}$ $\displaystyle\simeq\left(\mathbb{O\otimes
O}_{s}\right)H^{2}\simeq\left(\mathbb{O}\mathbb{\otimes
O}_{s}\right)\widetilde{H}^{2}.$ (51)

Again, there are two real forms of (49), namely the locally symmetric, pseudo-
Riemannian coset manifolds

$X_{128,I}:=\frac{\text{E}_{8(8)}}{\text{SO}^{\ast}\left(16\right)},\leavevmode\nobreak\
\\#_{nc}=72,\\#_{c}=56\Rightarrow\chi=16;$ (52)
$X_{128,II}:=\frac{\text{E}_{8(-24)}}{\text{SO}^{\ast}\left(16\right)},\leavevmode\nobreak\
\\#_{nc}=56,\\#_{c}=72\Rightarrow\chi=-16,$ (53)

whose isotropy Lie group has the corresponding Lie algebra which is not an
entry of any real form of the Magic Square of order 2.

In other words, since the Lie algebra $\mathfrak{so}^{\ast}(16)$ does not
occur in any real form of the Magic Square of order 2 (see Secs. 2.5 and 2.6),
the symmetric manifolds (52)-(53) cannot seemingly be characterized as
Rosenfeld planes over $\mathbb{O}$(or$\leavevmode\nobreak\
\mathbb{O}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$).
It should however be noticed that the isotropy group SO${}^{\ast}(16)$ of
(52)-(53) admits an interpretation in terms of the Magic Square of order 4,
namely

$X_{128,I}\simeq\frac{\mathcal{M}_{3}\left(\mathbb{O}_{s},\mathbb{O}_{s}\right)}{\mathcal{M}_{4}\left(\mathbb{H},\mathbb{H}_{s}\right)};$
(54)
$X_{128,II}\simeq\frac{\mathcal{M}_{3}\left(\mathbb{O},\mathbb{O}_{s}\right)}{\mathcal{M}_{4}\left(\mathbb{H},\mathbb{H}_{s}\right)},$
(55)

but still the rationale (if any) of such formulæ is missing, and it is surely
not the one underlying the “magic” formulæ (28)-(30).

Figure 2: _Cartan classification, Satake diagram, character $\chi$ of
exceptional Lie groups $\text{F}_{4}$ and $\text{E}_{6}$ and related
octonionic projective or hyperbolic Rosenfeld plane of which they are isometry
group._ Figure 3: _Cartan classification, Satake diagram, character $\chi$ of
exceptional Lie groups $\text{E}_{7}$ and $\text{E}_{8}$ and related
octonionic projective or hyperbolic Rosenfeld plane of which they are isometry
group._

##  5 “Magic” formulæ for Rosenfeld lines

Let Spin$\left(\mathbb{A}\right)$ the spin covering Lie group whose Lie
algebra is
Lie$\left(\text{Spin}\left(\mathbb{A}\right)\right)=\mathfrak{so}\left(\mathbb{A}\right)$,
namely the Lie algebra which preserves the norm of the Hurwitz algebra
$\mathbb{A}$. In order to characterize “projective” Rosenfeld lines over
tensor products of Hurwitz algebras in terms of coset manifolds with isometry
and isotropy Lie groups whose Lie algebras respectively are entries of real
forms of the Magic Square of order 2 and norm-preserving Lie algebras of the
involved Hurwitz algebras, we now introduce a variation of (28) and (29),
giving rise to the following two different classes of locally symmetric,
(pseudo-)Riemannian coset manifolds, that we name as Rosenfeld lines :

  1. 1.

The “projective” Rosenfeld line

$\left(\mathbb{A}\otimes\mathbb{B}\right)P^{1}\simeq\frac{\mathcal{M}_{2}\left(\mathbb{A},\mathbb{B}\right)}{\text{Spin}\left(\mathbb{A}\right)\otimes\text{Spin}\left(\mathbb{B}\right)}.$
(56)

  2. 2.

The hyperbolic Rosenfeld line

$\left(\mathbb{A}\otimes\mathbb{B}\right)H^{2}\simeq\frac{\mathcal{M}_{1,1}\left(\mathbb{A},\mathbb{B}\right)}{\text{Spin}\left(\mathbb{A}\right)\otimes\text{Spin}\left(\mathbb{B}\right)}.$
(57)

Eqs. (56) and (57) are named “magic” formulæ, since they characterize the
Rosenfeld lines as homogeneous (symmetric) manifolds, with isometry and
isotropy Lie groups respectively given by the entries of some real forms of
the Magic Square of order 2 and by norm-preserving Lie algebras of the
involved Hurwitz algebras.

##  6 Octonionic Rosenfeld lines

We now consider the “magic” formulæ (56)-(57) in the cases232323Also in this
case, this restriction does not imply any loss of generality, as far as the
other, non-octonionic Rosenfeld lines can be obtained as suitable sub-
manifolds of the octonionic Rosenfeld lines. in which $\mathbb{A}$ and/or
$\mathbb{B}$ is $\mathbb{O}$ or $\mathbb{O}_{s}$ : this will allow us to
rigorously introduce the octonionic Rosenfeld lines; however, again, it is
here worth anticipating that a few real forms of octonionic Rosenfeld lines
cannot be characterized in this way.

###  6.1 $\mathbb{O}\simeq\mathbb{R}\otimes\mathbb{O}$

Within the framework introduced above, the starting point is given by the
octonionic projective line over $\mathbb{C}$, which is nothing but the
8-sphere $S_{\mathbb{C}}^{8}$ over $\mathbb{C}$, i.e. the locally symmetric
coset manifold having as isometry group Spin$\left(9,\mathbb{C}\right)$ and as
isotropy group $\text{Spin}\left(9,\mathbb{C}\right)$ :

$\mathbb{O}P_{\mathbb{C}}^{1}\simeq\frac{\text{Spin}\left(9,\mathbb{C}\right)}{\text{Spin}\left(8,\mathbb{C}\right)}.$
(58)

In this coset space formulation, the tangent space of
$\mathbb{O}P_{\mathbb{C}}^{1}$ can be identified with the
$\mathbf{8}_{(v,s,c),\mathbb{C}}$ representation of
$\text{Spin}\left(8,\mathbb{C}\right)$ where $\mathfrak{d}_{4}$-triality [Por]
allows to equivalently choose $v$, $s$, or $c$.

Then, by specifying the formulæ (56) and (57) for $\mathbb{A}=\mathbb{R}$ and
$\mathbb{B}=\mathbb{O}$ or $\mathbb{O}_{s}$, we obtain all real forms242424By
real forms of $\mathbb{O}P_{\mathbb{C}}^{1}$, we here mean the cosets with
isometry groups given by all real (compact and non-compact) forms of
Spin$\left(9,\mathbb{C}\right)$, and with isotropy group given by all (compact
and non-compact) real forms of Spin$\left(8,\mathbb{C}\right)$ which are
subgroups of the corresponding real form of Spin$\left(9,\mathbb{C}\right)$.
of (58) which can be expressed as Rosenfeld lines over
$\mathbb{R}\otimes\mathbb{O}\simeq\mathbb{O}$ or over
$\mathbb{R}\otimes\mathbb{O}_{s}\simeq\mathbb{O}_{s}$; they are summarized by
the following Table :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\mathbb{O}P^{1}$ | $\text{Spin}\left(9\right)$ | $\text{Spin}\left(8\right)$ | $0$ | $8$ | $-8$
$\mathbb{O}H^{1}$ | $\text{Spin}\left(8,1\right)$ | $\text{Spin}\left(8\right)$ | $8$ | $0$ | $8$
$\mathbb{O}_{s}P^{1}$ | $\text{Spin}\left(5,4\right)$ | $\text{Spin}\left(4,4\right)$ | $4$ | $4$ | $0$

along with $\mathbb{O}_{s}H^{1}\simeq\mathbb{O}_{s}P^{1}$.

It should then be remarked that the octonionic line $\mathbb{O}P^{1}$ can be
identified with the 8-sphere $S^{8}\equiv S_{\mathbb{R}}^{8}$ over
$\mathbb{R}$, the hyperbolic line $\mathbb{O}H^{1}$ with the 8-hyperboloid
$H^{8}$ and the split-octonionic line $\mathbb{O}_{s}P^{1}$ with the Kleinian
8-hyperboloid $H^{4,4}$ (for some applications to physics and to non-compact
versions of Hopf maps, see e.g. [Ha]).

The real forms of $\mathbb{O}P_{\mathbb{C}}^{1}$ (58) have the general
structures

$\underset{\text{structure\leavevmode\nobreak\
}I}{\frac{\text{Spin}\left(p,9-p\right)}{\text{Spin}\left(p,8-p\right)}}\leavevmode\nobreak\
\,\,\text{ or \,\,}\leavevmode\nobreak\
\underset{\text{structure\leavevmode\nobreak\
}II}{\frac{\text{Spin}\left(p,9-p\right)}{\text{Spin}\left(p-1,9-p\right)}},$
(59)

with $p=0,1,...,9$, which gets exchanged under $p\leftrightarrow 9-p$.

$\mathbb{O}P^{1}$, $\mathbb{O}H^{1}$ and
$\mathbb{O}_{s}P^{1}\simeq\mathbb{O}_{s}H^{1}$ respectively correspond to
structure $I$ with $p=0$, $8$, $4$ (or structure $II$ with $p=9$, $1$, $5$).

All other real forms of $\mathbb{O}P_{\mathbb{C}}^{1}$ (58) cannot be
characterized as Rosenfeld lines over $\mathbb{O}$ (or$\leavevmode\nobreak\
\mathbb{O}_{s}$).

###  6.2 $\mathbb{C}\otimes\mathbb{O}$

In the bioctonionic case, i.e. for $\mathbb{C}\otimes\mathbb{O}$, the starting
point is given by the bioctonionic projective line over $\mathbb{C}$, i.e. by
the locally symmetric coset manifold having as isometry group
$\text{Spin}\left(10,\mathbb{C}\right)$ with
$\text{Spin}\left(8,\mathbb{C}\right)\otimes$U$\left(1\right)$ as isotropy
group :

$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}\simeq\frac{\text{Spin}\left(10,\mathbb{C}\right)}{\text{Spin}\left(8,\mathbb{C}\right)\otimes\left(\text{U}\left(1\right)\right)_{\mathbb{C}}}.$
(60)

In this coset space formulation, the tangent space of
$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$ can be identified
with the
$\mathbf{8}_{(v,s,c),\mathbb{C}+}\oplus\mathbf{8}_{(v,s,c),\mathbb{C}-}$
representation of
$\text{Spin}\left(8,\mathbb{C}\right)\otimes\left(\text{U}\left(1\right)\right)_{\mathbb{C}}$,
where, as above, $\mathfrak{d}_{4}$-triality [Por] allows to equivalently
choose $v$, $s$, or $c$.

Then, by specifying the formulæ (56) and (57) for $\mathbb{A}=\mathbb{C}$ or
$\mathbb{C}_{s}$ and $\mathbb{B}=\mathbb{O}$ or $\mathbb{O}_{s}$, we obtain
all real forms252525By real forms of
$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$, we here mean the
cosets with isometry groups given by all real (compact and non-compact) forms
of Spin$\left(10,\mathbb{C}\right)$, and with isotropy group given by all
(compact and non-compact) real forms of
Spin$\left(8,\mathbb{C}\right)\otimes\left(\text{U}\left(1\right)\right)_{\mathbb{C}}$
which are subgroups of the corresponding real form of
Spin$\left(10,\mathbb{C}\right)$. of (60) which can be expressed as Rosenfeld
lines over $\mathbb{C}$(or$\leavevmode\nobreak\
\mathbb{C}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$);
they are summarized by the following Table :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\left(\mathbb{C}\otimes\mathbb{O}\right)P^{1}$ | $\text{Spin}\left(10\right)$ | $\text{Spin}\left(8\right)\otimes\text{U}\left(1\right)$ | $0$ | $16$ | $-16$
$\left(\mathbb{C}\otimes\mathbb{O}\right)H^{1}$ | $\text{Spin}\left(8,2\right)$ | $\text{Spin}\left(8\right)\otimes\text{U}\left(1\right)$ | $16$ | $0$ | $16$
$\left(\mathbb{C}\otimes\mathbb{O}_{s}\right)P^{1}$ | $\text{Spin}\left(6,4\right)$ | $\text{Spin}\left(4,4\right)\otimes\text{U}\left(1\right)$ | $8$ | $8$ | $0$
$\left(\mathbb{C}_{s}\otimes\mathbb{O}\right)P^{1}$ | $\text{Spin}\left(9,1\right)$ | $\text{Spin}\left(8\right)\otimes\text{SO}\left(1,1\right)$ | $8$ | $8$ | $0$
$\left(\mathbb{C}_{s}\otimes\mathbb{O}_{s}\right)P^{1}$ | $\text{Spin}\left(5,5\right)$ | $\text{Spin}\left(4,4\right)\otimes\text{SO}\left(1,1\right)$ | $8$ | $8$ | $0$

along with

$\displaystyle\left(\mathbb{C\otimes O}_{s}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{C\otimes O}_{s}\right)H^{1};$ (61)
$\displaystyle\left(\mathbb{C}_{s}\mathbb{\otimes O}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{C}_{s}\mathbb{\otimes O}\right)H^{1};$ (62)
$\displaystyle\left(\mathbb{C}_{s}\mathbb{\otimes O}_{s}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{C}_{s}\mathbb{\otimes O}_{s}\right)H^{1}.$
(63)

The real forms of $\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$
(60) have the general structures

$\displaystyle\underset{\text{structure\leavevmode\nobreak\
}I}{\frac{\text{Spin}\left(p,10-p;\mathbb{C}\right)}{\text{Spin}\left(p,8-p;\mathbb{C}\right)\otimes
U_{1}}\,}\,\text{\,\,\, or
}\,\,\,\,\underset{\text{structure\leavevmode\nobreak\
}II}{\frac{\text{Spin}\left(p,10-p;\mathbb{C}\right)}{\text{Spin}\left(p-2,10-p;\mathbb{C}\right)\otimes
U_{1}}},$ (64) $\displaystyle\underset{\text{structure\leavevmode\nobreak\
}III}{\frac{\text{Spin}\left(p,10-p;\mathbb{C}\right)}{\text{Spin}\left(p-1,9-p;\mathbb{C}\right)\otimes
SO\left(1,1\right)}},$ (65) $\displaystyle
Y_{16}:=\frac{\text{SO}^{\ast}\left(10\right)}{\text{SO}^{\ast}\left(8\right)\otimes\text{U}\left(1\right)}\simeq\frac{\text{SO}^{\ast}\left(10\right)}{\text{Spin}\left(6,2\right)\otimes\text{U}\left(1\right)},$
(66)

with $p=0,1,...,10$, such that structures $I$ and $II$ get exchanged (while
structure $III$ is invariant) under $p\leftrightarrow 10-p$.

The planes $\left(\mathbb{C\otimes O}\right)P^{1}$, $\left(\mathbb{C\otimes
O}\right)H^{1}$, $\left(\mathbb{C\otimes O}_{s}\right)P^{1}$ respectively
correspond to structure $I$ with $p=0$, $8$, $6$ (or structure $II$ with
$p=10$, $2$, $4$), whereas $\left(\mathbb{C}_{s}\mathbb{\otimes
O}\right)P^{1}$ and $\left(\mathbb{C}_{s}\mathbb{\otimes O}_{s}\right)P^{1}$
respectively correspond to structure $III$ with $p=9$ (or $p=1$) and $p=5$.
All other real forms of
$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$ (60), and in
particular (66), cannot be characterized as Rosenfeld lines over
$\mathbb{C}$(or$\leavevmode\nobreak\
\mathbb{C}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$).

###  6.3 $\mathbb{H}\otimes\mathbb{O}$

In the quateroctonionic case, i.e. for $\mathbb{H}\otimes\mathbb{O}$, the
starting point is given by the quateroctonionic projective line over
$\mathbb{C}$, i.e. by the locally symmetric coset manifold having as isometry
group $\text{Spin}\left(12,\mathbb{C}\right)$ with
$\text{Spin}\left(8,\mathbb{C}\right)\otimes$Spin$\left(4,\mathbb{C}\right)$
as isotropy group :

$\left(\mathbb{H}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}\simeq\frac{\text{Spin}\left(12,\mathbb{C}\right)}{\text{Spin}\left(8,\mathbb{C}\right)\otimes\text{Spin}\left(4,\mathbb{C}\right)}.$
(67)

In this coset space formulation, the tangent space of
$\left(\mathbb{H}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$ can be identified
with the
$\left(\boldsymbol{8}_{\left(v,s,c\right)},\boldsymbol{2},\boldsymbol{2}\right)$
representation of
Spin$\left(8,\mathbb{C}\right)\otimes$Spin$\left(4,\mathbb{C}\right)$, where
we used
Spin$\left(4,\mathbb{C}\right)\simeq$SL$\left(2,\mathbb{C}\right)\otimes$SL$\left(2,\mathbb{C}\right)$
and, as above, $\mathfrak{d}_{4}$-triality [Por] allows to equivalently choose
$v$, $s$, or $c$.

Then, by specifying the formulæ (56) and (57) for $\mathbb{A}=\mathbb{H}$ or
$\mathbb{H}_{s}$ and $\mathbb{B}=\mathbb{O}$ or $\mathbb{O}_{s}$, we obtain
all real forms262626By real forms of
$\left(\mathbb{H}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$, we here mean the
cosets with isometry groups given by all real (compact and non-compact) forms
of Spin$\left(12,\mathbb{C}\right)$, and with isotropy group given by all
(compact and non-compact) real forms of
Spin$\left(8,\mathbb{C}\right)\otimes$Spin$\left(4,\mathbb{C}\right)$ which
are subgroups of the corresponding real form of
Spin$\left(12,\mathbb{C}\right)$. of (67) which can be expressed as Rosenfeld
lines over $\mathbb{H}$(or$\leavevmode\nobreak\
\mathbb{H}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$);
they are summarized by the following Table :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\left(\mathbb{H}\otimes\mathbb{O}\right)P^{1}$ | $\text{Spin}\left(12\right)$ | $\text{Spin}\left(8\right)\otimes\text{SU}\left(2\right)\otimes\text{SU}\left(2\right)$ | $0$ | $32$ | $-32$
$\left(\mathbb{H}\otimes\mathbb{O}\right)H^{1}$ | $\text{Spin}\left(8,4\right)$ | $\text{Spin}\left(8\right)\otimes\text{SU}\left(2\right)\otimes\text{SU}\left(2\right)$ | $32$ | $0$ | $32$
$\left(\mathbb{H}\otimes\mathbb{O}_{s}\right)P^{1}$ | $\text{Spin}\left(8,4\right)$ | $\text{Spin}\left(4,4\right)\otimes\text{SU}\left(2\right)\otimes\text{SU}\left(2\right)$ | $16$ | $16$ | $0$
$\left(\mathbb{H}_{s}\otimes\mathbb{O}\right)P^{1}$ | $\text{Spin}\left(10,2\right)$ | $\text{Spin}\left(8\right)\otimes\text{SL}\left(2,\mathbb{R}\right)\otimes\text{SL}\left(2,\mathbb{R}\right)$ | $16$ | $16$ | $0$
$\left(\mathbb{H}_{s}\otimes\mathbb{O}_{s}\right)P^{1}$ | $\text{Spin}\left(6,6\right)$ | $\text{Spin}\left(4,4\right)\otimes\text{SL}\left(2,\mathbb{R}\right)\otimes\text{SL}\left(2,\mathbb{R}\right)$ | $16$ | $16$ | $0$

along with

$\displaystyle\left(\mathbb{H\otimes O}_{s}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{H\otimes O}_{s}\right)H^{1};$ (68)
$\displaystyle\left(\mathbb{H}_{s}\mathbb{\otimes O}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{H}_{s}\mathbb{\otimes O}\right)H^{1};$ (69)
$\displaystyle\left(\mathbb{H}_{s}\mathbb{\otimes O}_{s}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{H}_{s}\mathbb{\otimes O}_{s}\right)H^{1}.$
(70)

The real forms of $\left(\mathbb{H}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$
(67) have the general structures

$\displaystyle\underset{\text{structure\leavevmode\nobreak\
}I}{\frac{\text{Spin}\left(p,12-p;\mathbb{C}\right)}{\text{Spin}\left(p,8-p;\mathbb{C}\right)\otimes
SU(2)^{\otimes 2}}}\,\,\,\leavevmode\nobreak\
\text{or}\,\,\,\,\leavevmode\nobreak\
\underset{\text{structure\leavevmode\nobreak\
}II}{\frac{\text{Spin}\left(p,12-p;\mathbb{C}\right)}{\text{Spin}\left(p-4,12-p;\mathbb{C}\right)\otimes
SU(2)_{1}^{\otimes 2}}},$ (71)
$\displaystyle\underset{\text{structure\leavevmode\nobreak\
}III}{\frac{\text{Spin}\left(p,12-p;\mathbb{C}\right)}{\text{Spin}\left(p-1,9-p;\mathbb{C}\right)\otimes
SL(2,\mathbb{C})_{\mathbb{R}}}}\leavevmode\nobreak\
\,\,\,\text{or}\leavevmode\nobreak\
\,\,\,\,\underset{\text{structure\leavevmode\nobreak\
}IV}{\frac{\text{Spin}\left(p,12-p;\mathbb{C}\right)}{\text{Spin}\left(p-3,11-p;\mathbb{C}\right)\otimes
SL(2,\mathbb{C})_{\mathbb{R}}}},$ (72)
$\displaystyle\underset{\text{structure\leavevmode\nobreak\
}V}{\frac{\text{Spin}\left(p,12-p;\mathbb{C}\right)}{\text{Spin}\left(p-2,10-p;\mathbb{C}\right)\otimes
SL(2,\mathbb{R})^{\otimes 2}}},$ (73) $\displaystyle
Y_{32}:=\frac{\text{SO}^{\ast}\left(12\right)}{\text{SO}^{\ast}\left(8\right)\otimes\text{SO}^{\ast}\left(4\right)}\simeq\frac{\text{SO}^{\ast}\left(12\right)}{\text{Spin}\left(6,2\right)\otimes\text{SU}\left(2\right)\otimes\text{SL}\left(2,\mathbb{R}\right)},$
(74)

with $p=0,1,...,12$, such that structures $I$ and $II$ (and structures $III$
and $IV$) get exchanged (while structure $V$ is invariant) under
$p\leftrightarrow 12-p$.

$\left(\mathbb{H\otimes O}\right)P^{1}$, $\left(\mathbb{H\otimes
O}\right)H^{1}$ and $\left(\mathbb{H\otimes O}_{s}\right)P^{1}$ respectively
correspond to structure $I$ with $p=0$, $8$, $4$ (or structure $II$ with
$p=12$, $4$, $8$), whereas $\left(\mathbb{H}_{s}\mathbb{\otimes
O}\right)P^{1}$ and $\left(\mathbb{H}_{s}\mathbb{\otimes O}_{s}\right)P^{1}$
respectively correspond to structure $V$ with $p=2$ (or $p=10$) and $p=6$.

All other real forms of
$\left(\mathbb{H}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$ (67), and in
particular (74), cannot be characterized as Rosenfeld lines over
$\mathbb{H}$(or$\leavevmode\nobreak\
\mathbb{H}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$).

###  6.4 $\mathbb{O}\otimes\mathbb{O}$

In the octooctonionic case, i.e. for $\mathbb{O}\otimes\mathbb{O}$, the
starting point is given by the octooctonionic projective line over
$\mathbb{C}$, i.e. by the locally symmetric coset manifold having as isometry
group $\text{Spin}\left(16,\mathbb{C}\right)$ with
$\text{Spin}\left(8,\mathbb{C}\right)\otimes$Spin$\left(8,\mathbb{C}\right)$
as isotropy group :

$\left(\mathbb{O}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}\simeq\frac{\text{Spin}\left(12,\mathbb{C}\right)}{\text{Spin}\left(8,\mathbb{C}\right)\otimes\text{Spin}\left(8,\mathbb{C}\right)}.$
(75)

In this coset space formulation, the tangent space of
$\left(\mathbb{O}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$ can be identified
with the
$\left(\boldsymbol{8}_{\left(v,s,c\right)},\boldsymbol{8}_{\left(v,s,c\right)}\right)$
representation of
Spin$\left(8,\mathbb{C}\right)\otimes$Spin$\left(8,\mathbb{C}\right)$, where,
as above, $\mathfrak{d}_{4}$-triality [Por] allows to equivalently choose $v$,
$s$, or $c$ (in all possible pairs for the two Spin$\left(8,\mathbb{C}\right)$
factors of the isotropy group).

Then, by specifying the formulæ (56) and (57) for $\mathbb{A}=\mathbb{O}$ or
$\mathbb{O}_{s}$ and $\mathbb{B}=\mathbb{O}$ or $\mathbb{O}_{s}$, we obtain
all real forms272727By real forms of
$\left(\mathbb{O}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$, we here mean the
cosets with isometry groups given by all real (compact and non-compact) forms
of Spin$\left(16,\mathbb{C}\right)$, and with isotropy group given by all
(compact and non-compact) real forms of
Spin$\left(8,\mathbb{C}\right)\otimes$Spin$\left(8,\mathbb{C}\right)$ which
are subgroups of the corresponding real form of
Spin$\left(16,\mathbb{C}\right)$. of (75) which can be expressed as Rosenfeld
lines over $\mathbb{O}$(or$\leavevmode\nobreak\
\mathbb{O}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$);
they are summarized by the following Table :

Plane | Isometry | Isotropy | $\\#_{nc}$ | $\\#_{c}$ | $\chi$
---|---|---|---|---|---
$\left(\mathbb{O}\otimes\mathbb{O}\right)P^{1}$ | $\text{Spin}\left(16\right)$ | $\text{Spin}\left(8\right)\otimes\text{Spin}\left(8\right)$ | $0$ | $64$ | $-64$
$\left(\mathbb{O}\otimes\mathbb{O}\right)H^{1}$ | $\text{Spin}\left(8,8\right)$ | $\text{Spin}\left(8\right)\otimes\text{Spin}\left(8\right)$ | $64$ | $0$ | $64$
$\left(\mathbb{O}\otimes\mathbb{O}_{s}\right)P^{1}$ | $\text{Spin}\left(12,4\right)$ | $\text{Spin}\left(8\right)\otimes\text{Spin}\left(4,4\right)$ | $32$ | $32$ | $0$
$\left(\mathbb{O}_{s}\otimes\mathbb{O}_{s}\right)P^{1}$ | $\text{Spin}\left(8,8\right)$ | $\text{Spin}\left(4,4\right)\otimes\text{Spin}\left(4,4\right)$ | $32$ | $32$ | $0$

along with

$\displaystyle\left(\mathbb{O\otimes O}_{s}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{O\otimes O}_{s}\right)H^{1},$ (76)
$\displaystyle\left(\mathbb{O}_{s}\mathbb{\otimes O}_{s}\right)P^{1}$
$\displaystyle\simeq\left(\mathbb{O}_{s}\mathbb{\otimes O}_{s}\right)H^{1}.$
(77)

The real forms of $\left(\mathbb{O}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$
(75) have the general structures

$\displaystyle\underset{\text{structure\leavevmode\nobreak\
}I}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p,8-p;\mathbb{C}\right)\otimes\text{Spin}\left(8,\mathbb{C}\right)}}\leavevmode\nobreak\
\,\,\,\,\text{or}\,\,\,\,\leavevmode\nobreak\
\underset{\text{structure\leavevmode\nobreak\
}II}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-8,16-p;\mathbb{C}\right)\otimes\text{Spin}\left(8,\mathbb{C}\right)}},$
(78) $\displaystyle\underset{\text{structure\leavevmode\nobreak\
}III}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-1,9-p;\mathbb{C}\right)\otimes\text{Spin}\left(1,7;\mathbb{C}\right)}}\leavevmode\nobreak\
\text{\,\,\,or}\,\,\,\leavevmode\nobreak\
\underset{\text{structure\leavevmode\nobreak\
}IV}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-7,15-p;\mathbb{C}\right)\otimes\text{Spin}\left(1,7;\mathbb{C}\right)}},$
(79) $\displaystyle\underset{\text{structure\leavevmode\nobreak\
}V}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-2,10-p;\mathbb{C}\right)\otimes\text{Spin}\left(2,6;\mathbb{C}\right)}}\leavevmode\nobreak\
\text{\,\,\,or}\leavevmode\nobreak\
\,\,\,\underset{\text{structure\leavevmode\nobreak\
}VI}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-6,14-p;\mathbb{C}\right)\otimes\text{Spin}\left(2,6;\mathbb{C}\right)}},$
(80) $\displaystyle\underset{\text{structure\leavevmode\nobreak\
}VII}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-3,11-p;\mathbb{C}\right)\otimes\text{Spin}\left(3,5;\mathbb{C}\right)}}\leavevmode\nobreak\
\text{\,\,\,or}\leavevmode\nobreak\
\underset{\text{structure\leavevmode\nobreak\
}VIII}{\,\,\,\,\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-5,13-p;\mathbb{C}\right)\otimes\text{Spin}\left(3,5;\mathbb{C}\right)}},$
(81) $\displaystyle\underset{\text{structure\leavevmode\nobreak\
}IX}{\frac{\text{Spin}\left(p,16-p;\mathbb{C}\right)}{\text{Spin}\left(p-4,12-p;\mathbb{C}\right)\otimes\text{Spin}\left(4,4;\mathbb{C}\right)}},$
(82) $\displaystyle
Y_{64}:=\frac{\text{SO}^{\ast}\left(16\right)}{\text{SO}^{\ast}\left(8\right)\otimes\text{SO}^{\ast}\left(8\right)}\simeq\frac{\text{SO}^{\ast}\left(16\right)}{\text{Spin}\left(6,2\right)\otimes\text{Spin}\left(6,2\right)},$
(83)

with $p=0,1,...,16$, such that structures $I$ and $II$, $III$ and $IV$, $V$
and $VI$, $VII$ and $VIII$ get exchanged (while structure $IX$ is invariant)
under $p\leftrightarrow 16-p$.

The planes $\left(\mathbb{O\otimes O}\right)P^{1}$, $\left(\mathbb{O\otimes
O}\right)H^{1}$ and $\left(\mathbb{O\otimes O}_{s}\right)P^{1}$ respectively
correspond to structure $I$ with $p=0$, $8$, $4$ (or structure $II$ with
$p=16$, $8$, $12$), whereas $\left(\mathbb{O}_{s}\mathbb{\otimes
O}_{s}\right)P^{1}$ corresponds to structure $IX$ with $p=8$. All other real
forms of $\left(\mathbb{O}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{1}$ (75),
and in particular (83), cannot be characterized as Rosenfeld lines over
$\mathbb{O}$(or$\leavevmode\nobreak\
\mathbb{O}_{s}$)$\otimes\mathbb{O}$(or$\leavevmode\nobreak\ \mathbb{O}_{s}$).

##  7 Conclusions

In this work, in order to provide a rigorous characterization of Rosenfeld
“projective” spaces over (rank-2) tensor products of (division or split)
Hurwitz algebras, we have introduced some “magic” formulæ (28)-(30) (for
planes) and (56)-(57) (for lines), which allowed us to characterize many
Rosenfeld spaces as symmetric (pseudo-)Riemannian cosets, whose isometry and
isotropy groups have Lie algebras which are entries of real forms of the
order-3 (Freudenthal-Tits [Ti, Fr65]) Magic Square, or of the order-2 (Barton-
Sudbery [BS03]) Magic Square.

For “projective” planes, the application of the “magic” formulæ (28)-(30) to
the case in which at least one of the two Hurwitz algebras in the associated
tensor product is given by the octonions $\mathbb{O}$ or by the split
octonions $\mathbb{O}_{s}$, allows us to retrieve all (compact and non-
compact) real forms of the corresponding octonionic Rosenfeld planes, except
for a limited numbers of pseudo-Riemannian symmetric cosets, named $X_{32,I}$,
$X_{32,II}$, $X_{64,I}$, $X_{64,II}$ ,$X_{64,III}$, $X_{128,I}$ and
$X_{128,II}$, and respectively given by (37), (38), (43), (44), (45), (52) and
(53). All such cosets share a common property : up to some possible rank-1 Lie
group factor, their isotropy group is given by the non-compact Lie group
SO${}^{\ast}\left(N\right)$ for $N=10,12,16$. The fact that such spaces are
not encompassed by our “magic” formulæ (28)-(30) is ultimately due to the fact
that the corresponding Lie algebra $\mathfrak{so}^{\ast}(N)$ does not occur in
any real form of the Magic Square of order 2 (except for the case $N=8$, for
which it holds the special isomorphism
$\mathfrak{so}^{\ast}(8)\simeq\mathfrak{so}(6,2)$).

For “projective” lines, the application of the “magic” formulæ (56)-(57) to
the case in which at least one of the two Hurwitz algebras in the associated
tensor product is given by the octonions $\mathbb{O}$ or by the split
octonions $\mathbb{O}_{s}$, allows us to retrieve some (compact and non-
compact) real forms of the corresponding octonionic Rosenfeld lines, but still
a number of other real forms of Rosenfeld lines is left out. Again, there are
spaces, such as $Y_{16},Y_{32}$ and $Y_{64}$ (respectively given by (66), (74)
and (83)) that have their isometry and isotropy groups containing factors
SO${}^{\ast}(M)$ for $M=4,8,10,12,16$; but there are also other pseudo-
Riemannian spaces left out, given by (59), (64)-(65), (71)-(73) and (78)-(82).

The fact that some pseudo-Riemannian symmetric spaces are not covered by the
classification yielded by the “magic” formulæ (28)-(30) and (56)-(57), which
are related to the entries of the Magic Squares of order 2 and 3, means that
not all real forms of Rosenfeld spaces can be realized as “projective” lines
or planes over (rank 2) tensor products of (division or split) Hurwitz
algebras. If one still believe that Rosenfeld’s approach was right after all,
one might want to extend our “magic” formulæ to involve not only unital,
alternative composition algebras, i.e. (division or split) Hurwitz algebras,
but also other algebras which are not unital or alternative. Extensive work on
Magic Squares over flexible composition algebras has been done by Elduque (see
e.g. [El04, El02]), whereas a geometrical framework for these algebras has
been recently discussed in [CZ22] and [CMZ22].

Therefore, after all, Rosenfeld might still be right….we hope to report on
this in forthcoming investigations.

## Acknowledgments

The work of D. Corradetti is supported by a grant of the Quantum Gravity
Research Institute. The work of AM is supported by a “Maria Zambrano”
distinguished researcher fellowship at the University of Murcia, ES, financed
by the European Union within the NextGenerationEU program.

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