# On the Structure of Graded Lie Superalgebras

Antonio J. Calderón Martín
José M. Sánchez Delgado
Departamento de Matemáticas.
Universidad de Cádiz. 11510 Puerto Real, Cádiz, Spain.
e-mail<EMAIL_ADDRESS>
e-mail<EMAIL_ADDRESS>Supported by the PCI of the UCA
‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project
numbers FQM298, FQM2467, FQM3737 and by the project of the Spanish Ministerio
de Educación y Ciencia MTM2007-60333.

###### Abstract

We study the structure of graded Lie superalgebras with arbitrary dimension
and over an arbitrary field 𝕂. We show that any of such algebras
${\mathfrak{L}}$ with a symmetric $G$-support is of the form
${\mathfrak{L}}=U+\sum\limits_{j}I_{j}$ with $U$ a subspace of
${\mathfrak{L}}_{1}$ and any $I_{j}$ a well described graded ideal of
${\mathfrak{L}}$, satisfying $[I_{j},I_{k}]=0$ if $j\neq k$. Under certain
conditions, it is shown that ${\mathfrak{L}}=(\bigoplus\limits_{k\in
K}I_{k})\oplus(\bigoplus\limits_{q\in Q}I_{q}),$ where any $I_{k}$ is a gr-
simple graded ideal of ${\mathfrak{L}}$ and any $I_{q}$ a completely
determined low dimensional non gr-simple graded ideal of ${\mathfrak{L}}$,
satisfying $[I_{q},I_{q^{\prime}}]=0$ for any $q^{\prime}\in Q$ with $q\neq
q^{\prime}$.

Keywords: Graded Lie superalgebras, Infinite dimensional Lie superalgebras,
Structure Theory.

2000 MSC: 17B70, 17B65, 17B05.

##  1 Introduction and previous definitions

Throughout this paper, Lie superalgebras ${\mathfrak{L}}$ are considered of
arbitrary dimension and over an arbitrary field 𝕂. It is worth to mention
that, unless otherwise stated, there is not any restriction on
$\dim{\mathfrak{L}}_{g}$ or the products
$[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$, where ${\mathfrak{L}}_{g}$
denotes the homogeneous subspace associated to $g\in G$. Lie superalgebras
plays an important role in theoretical physics, specially in conformal field
theory and supersymmetries (see [4, 17, 21] for recent references). The notion
of supersymmetry reflects the known symmetry between bosons and fermions,
being the mathematical structure formalizing this idea the one of supergroup,
or ${\hbox{\hueca Z}}_{2}$-graded Lie group. As mentioned in [22], its job is
that of modelling continuous supersymmetry transformations between bosons and
fermions. As Lie algebras consist of generators of Lie groups, the
infinitesimal Lie group elements tangent to the identity, so ${\hbox{\hueca
Z}}_{2}$–graded Lie algebras, otherwise known as Lie superalgebras, consist of
generators of (or infinitesimal) supersymmetry transformations. We also refers
to [3] and [13] for more interesting applications of Lie superalgebras. The
interest on gradings on simple Lie algebras has been remarkable in the last
years. The gradings of finite dimensional simple Lie algebras, ruling out
$\mathfrak{a}_{l}$, $\mathfrak{d}_{4}$ and the exceptional cases, are
described in [1]. The fine gradings on $\mathfrak{a}_{l}$ have been determined
in [14] solving the related problem of finding maximal abelian groups of
diagonalizable automorphisms of the algebras (not only in ${\rm
GL}(n,\mathbb{C})$ but also in ${{\rm O}}(n,\mathbb{C})$ for $n\neq 8$ and
${{\rm SP}}(2n,\mathbb{C})$). See also [12, 16, 19, 20]. The first studies of
gradings on exceptional Lie algebras are [2], [10] and [11], which describe
the group gradings on $\hbox{\got g}_{2}$ and $\hbox{\got f}_{4}$. The study
of the gradings of the real forms of complex Lie algebras begins in [15],
where are considered the gradings on the real forms of classical simple
complex Lie algebras. The description of the fine gradings of the real forms
of the exceptional simple Lie algebras ${\mathfrak{f}}_{4}$ and
${\mathfrak{g}}_{2}$ are obtained in [7]. Respect to the group gradings on
superalgebras, these have been considered, for the case of the Jordan
superalgebra $K_{10}$, in [8].

In the present paper we begin the study of arbitrary graded Lie superalgebras,
(not necessarily simple or finite-dimensional), and over an arbitrary field 𝕂
by focussing on their structure. In $\S 2$ we extend the techniques of
connections in the support of $G$ developed for graded Lie algebras in [6] to
the framework of graded Lie superalgebras ${\mathfrak{L}}$, so as to show that
${\mathfrak{L}}$ is of the form ${\mathfrak{L}}=U+\sum\limits_{j}I_{j}$ with
$U$ a subspace of ${\mathfrak{L}}_{1}$ and any $I_{j}$ a well described graded
ideal of ${\mathfrak{L}}$, satisfying $[I_{j},I_{k}]=0$ if $j\neq k$. In $\S
3$, and under certain conditions, it is shown that
${\mathfrak{L}}=(\bigoplus\limits_{k\in K}I_{k})\oplus(\bigoplus\limits_{q\in
Q}I_{q}),$ where any $I_{k}$ is a gr-simple graded ideal of ${\mathfrak{L}}$
and any $I_{q}$ a completely determined low dimensional non gr-simple graded
ideal of ${\mathfrak{L}}$, satisfying $[I_{q},I_{q^{\prime}}]=0$ for any
$q^{\prime}\in Q$ with $q\neq q^{\prime}$.

A Lie superalgebra ${\mathfrak{L}}$ is a ${\hbox{\hueca Z}}_{2}$-graded
algebra
${\mathfrak{L}}={\mathfrak{L}}^{\bar{0}}\oplus{\mathfrak{L}}^{\bar{1}}$ over
an arbitrary ground field 𝕂, endowed with a bilinear product $[\cdot,\cdot]$,
whose homogenous elements
$x\in{\mathfrak{L}}^{\bar{i}},y\in{\mathfrak{L}}^{\bar{j}},{\bar{i}},{\bar{j}}\in{\hbox{\hueca
Z}}_{2}$, satisfy

$[x,y]\subset{\mathfrak{L}}^{{\bar{i}}+{\bar{j}}}$
$[x,y]=-(-1)^{{\bar{i}}{\bar{j}}}[y,x]\hbox{ (Skew-supersymmetry)}$
$[x,[y,z]]=[[x,y],z]+(-1)^{{\bar{i}}{\bar{j}}}[y,[x,z]]\hbox{ (Super Jacobi
identity)}$

for any homogenous element $z\in{\mathfrak{L}}^{\bar{k}}$,
${\bar{k}}\in{\hbox{\hueca Z}}_{2}$.

Note that if ${\rm char}({\hbox{\hueca K}})\neq 2$, then
${\mathfrak{L}}^{\bar{0}}$ is a Lie algebra called the even or bosonic part of
${\mathfrak{L}}$ while ${\mathfrak{L}}^{\bar{1}}$ is called the odd or
fermonic part of ${\mathfrak{L}}$.

The term grading will always mean abelian group grading compatible with the
${\hbox{\hueca Z}}_{2}$-graduation providing the superalgebra structure of
${\mathfrak{L}}$. That is, a decomposition in vector subspaces

${\mathfrak{L}}=\bigoplus\limits_{g\in G}{\mathfrak{L}}_{g}$

where $G$ is an abelian group, and the homogeneous subspaces satisfy

${\mathfrak{L}}_{g}={\mathfrak{L}}_{g}^{\bar{0}}\oplus{\mathfrak{L}}_{g}^{\bar{1}}$
with
${\mathfrak{L}}_{g}^{\bar{i}}={\mathfrak{L}}_{g}\cap{\mathfrak{L}}^{\bar{i}},{\bar{i}}\in{\hbox{\hueca
Z}}_{2}$, (1)

and

$[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{\prime}}]\subset{\mathfrak{L}}_{gg^{\prime}},$

(denoting by juxtaposition the product in $G$). We note that a $G$-grading of
${\mathfrak{L}}$ provides a refinement of the initial ${\hbox{\hueca
Z}}_{2}$-grading of ${\mathfrak{L}}$ (see [18, Definition 3.1.4]), and that
split Lie superalgebras and graded Lie algebras are examples of graded Lie
superalgebras. Hence, the present paper extends the results in [6, 9].

We call the $G$-support of the grading to the set

$\Sigma_{G}:=\\{g\in G\setminus\\{1\\}:{\mathfrak{L}}_{g}\neq 0\\}.$

So we can write

${\mathfrak{L}}=\bigoplus\limits_{g\in
G}({\mathfrak{L}}_{g}^{\bar{0}}\oplus{\mathfrak{L}}_{g}^{\bar{1}})=({\mathfrak{L}}_{1}^{\bar{0}}\oplus{\mathfrak{L}}_{1}^{\bar{1}})\oplus(\bigoplus\limits_{g\in\Sigma_{G}}({\mathfrak{L}}_{g}^{\bar{0}}\oplus{\mathfrak{L}}_{g}^{\bar{1}}))$.

We also denote by
$\Sigma_{G}^{\bar{0}}:=\\{g\in\Sigma_{G}:{\mathfrak{L}}_{g}^{\bar{0}}\neq
0\\}$ and by
$\Sigma_{G}^{\bar{1}}:=\\{g\in\Sigma_{G}:{\mathfrak{L}}_{g}^{\bar{1}}\neq
0\\}$. So $\Sigma_{G}=\Sigma_{G}^{\bar{0}}\cup\Sigma_{G}^{\bar{1}}$, being a
non necessarily disjoint union.

The $G$-support $\Sigma_{G}$ is called symmetric if $g\in\Sigma_{G}^{\bar{i}}$
implies $g^{-1}\in\Sigma_{G}^{\bar{i}},{\bar{i}}\in{\hbox{\hueca Z}}_{2}$.

The usual regularity concepts will be understood in the graded sense,
(compatible with the initial ${\hbox{\hueca Z}}_{2}$-graduation of
${\mathfrak{L}}$). That is, a graded ideal $I$ of ${\mathfrak{L}}$ is an ideal
which splits as

$I=\bigoplus\limits_{g\in G}I_{g}=\bigoplus\limits_{g\in
G}(I_{g}^{\bar{0}}\oplus I_{g}^{\bar{1}})$ with any
$I_{g}^{\bar{i}}=I_{g}\cap{\mathfrak{L}}^{\bar{i}},{\bar{i}}\in{\hbox{\hueca
Z}}_{2}$. (2)

A graded Lie superalgebra ${\mathfrak{L}}$ will be called gr-simple if
$[{\mathfrak{L}},{\mathfrak{L}}]\neq 0$ and its only graded ideals are
$\\{0\\}$ and ${\mathfrak{L}}$.

Observe that from the grading of ${\mathfrak{L}}$ and equation (1) we get

$[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{\prime}}^{\bar{j}}]\subset{\mathfrak{L}}_{gg^{\prime}}^{{\bar{i}}+{\bar{j}}}$

for any ${\bar{i}},{\bar{j}}\in{\hbox{\hueca Z}}_{2}$.

##  2 Connections in $\Sigma_{G}$. Decompositions

From now on and throughout the paper, ${\mathfrak{L}}$ denotes a graded Lie
superalgebra with a symmetric $G$-support $\Sigma_{G}$, and

${\mathfrak{L}}=\bigoplus\limits_{g\in
G}({\mathfrak{L}}_{g}^{\bar{0}}\oplus{\mathfrak{L}}_{g}^{\bar{1}})=({\mathfrak{L}}_{1}^{\bar{0}}\oplus{\mathfrak{L}}_{1}^{\bar{1}})\oplus(\bigoplus\limits_{g\in\Sigma_{G}}({\mathfrak{L}}_{g}^{\bar{0}}\oplus{\mathfrak{L}}_{g}^{\bar{1}}))$

the corresponding grading. We begin by developing connection techniques in
this framework.

######  Definition 2.1.

Let $g$ and $g^{\prime}$ be two elements in $\Sigma_{G}$. We shall say that
$g$ is $\Sigma_{G}$-connected to $g^{\prime}$ if there exist
$g_{1},g_{2}...,g_{n}\in\Sigma_{G}$ such that

  1. 1.

$g_{1}=g$.

  2. 2.

$\\{g_{1},g_{1}g_{2},......,g_{1}g_{2}\cdots g_{n-1}\\}\subset\Sigma_{G}.$

  3. 3.

$g_{1}g_{2}\cdots g_{n}\in\\{g^{\prime},{(g^{\prime})}^{-1}\\}$.

We also say that $\\{g_{1},...,g_{n}\\}$ is a $\Sigma_{G}$-connection from $g$
to $g^{\prime}$.

The next result shows the $\Sigma_{G}$-connection relation is of equivalence.
Its proof is virtually identical to the proof of [6, Proposition 2.1] but for
completeness reasons we add an sketch of the same.

######  Proposition 2.1.

The relation $\sim$ in $\Sigma_{G}$, defined by $g\sim g^{\prime}$ if and only
if $g$ is $\Sigma_{G}$-connected to $g$, is of equivalence.

###### Proof.

$\\{g\\}$ is a $\Sigma_{G}$-connection from $g$ to itself and therefore $g\sim
g$.

If $g\sim g^{\prime}$ and $\\{g_{1},...,g_{n}\\}$ is a $\Sigma_{G}$-connection
from $g$ to $g^{\prime}$, then

$\\{g_{1}\cdots g_{n},g_{n}^{-1},g_{n-1}^{-1},...,g_{2}^{-1}\\}$

is a $\Sigma_{G}$-connection from $g^{\prime}$ to $g$ in case $g_{1}\cdots
g_{n}=g^{\prime}$, and

$\\{g_{1}^{-1}\cdots g_{n}^{-1},g_{n},g_{n-1},...,g_{2}\\}$

in case $g_{1}\cdots g_{n}=(g^{\prime})^{-1}$. Therefore $g^{\prime}\sim g$.

Finally, suppose $g\sim g^{\prime}$ and $g^{\prime}\sim g^{\prime\prime}$, and
write $\\{g_{1},...,g_{n}\\}$ for a $\Sigma_{G}$-connection from $g$ to
$g^{\prime}$ and $\\{{g^{\prime}}_{1},...,{g^{\prime}}_{m}\\}$ for a
$\Sigma_{G}$-connection from $g^{\prime}$ to $g^{\prime\prime}$. If $m>1$,
then $\\{g_{1},...,g_{n},{g^{\prime}}_{2},...,{g^{\prime}}_{m}\\}$ is a
$\Sigma_{G}$-connection from $g$ to $g^{\prime\prime}$ in case $g_{1}\cdots
g_{n}=g^{\prime}$, and
$\\{g_{1},...,g_{n},{g^{\prime}}_{2}^{-1},...,{g^{\prime}}_{m}^{-1}\\}$ in
case $g_{1}\cdots g_{n}=(g^{\prime})^{-1}$. If $m=1$, then
$g^{\prime\prime}\in\\{g^{\prime},(g^{\prime})^{-1}\\}$ and so
$\\{g_{1},...,g_{n}\\}$ is a $\Sigma_{G}$-connection from $g$ to
$g^{\prime\prime}$. Therefore $g\sim g^{\prime\prime}$ and $\sim$ is of
equivalence. ∎

Given $g\in\Sigma_{g}$, we denote by

${\mathfrak{C}}_{g}:=\\{g^{\prime}\in\Sigma_{G}:g\hskip 2.84544pt{\rm
is}\hskip 2.84544pt\Sigma_{G}{\rm-connected}\hskip 2.84544pt{\rm to}\hskip
2.84544ptg^{\prime}\\}.$

Clearly if $g^{\prime}\in{\mathfrak{C}}_{g}$ then
${(g^{\prime})}^{-1}\in{\mathfrak{C}}_{g}$ and, by Proposition 2.1, if
$h\notin{\mathfrak{C}}_{g}$ then
${\mathfrak{C}}_{g}\cap{\mathfrak{C}}_{h}=\emptyset$.

######  Lemma 2.1.

If $g^{\prime}\in{\mathfrak{C}}_{g}$ and
$g^{\prime\prime},g^{\prime}g^{\prime\prime}\in\Sigma_{G}$, then
$g^{\prime\prime},g^{\prime}g^{\prime\prime}\in{\mathfrak{C}}_{g}$.

###### Proof.

The $\Sigma_{G}$-connection $\\{g^{\prime},g^{\prime\prime}\\}$ gives us
$g^{\prime}\sim g^{\prime}g^{\prime\prime}$. Hence, by the transitivity of
$\sim$, we finally get $g^{\prime}g^{\prime\prime}\in{\mathfrak{C}}_{g}$. To
verify $g^{\prime\prime}\in{\mathfrak{C}}_{g}$, observe that
$\\{g^{\prime}g^{\prime\prime},(g^{\prime})^{-1}\\}$ is a
$\Sigma_{G}$-connection from $g^{\prime}g^{\prime\prime}$ to
$g^{\prime\prime}$. Now, taking into account
$g^{\prime}g^{\prime\prime}\in{\mathfrak{C}}_{g}$, we conclude as above that
$g^{\prime\prime}\in{\mathfrak{C}}_{g}$. ∎

Our next goal is to associate an (adequate) graded ideal $I_{[g]}$ to any
${\mathfrak{C}}_{g}$. For ${\mathfrak{C}}_{g}$, $g\in\Sigma_{G}$, we define

${\mathfrak{L}}_{{\mathfrak{C}}_{g},1}:=span_{\hbox{\hueca
K}}\\{[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}]:g^{\prime}\in{\mathfrak{C}}_{g}\\}=$
$(\sum\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}([{\mathfrak{L}}_{g^{\prime}}^{\bar{0}},{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{0}}]+[{\mathfrak{L}}_{g^{\prime}}^{\bar{1}},{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{1}}])\oplus(\sum\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}[{\mathfrak{L}}_{g^{\prime}}^{\bar{0}},{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{1}}]+[{\mathfrak{L}}_{g^{\prime}}^{\bar{1}},{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{0}}]))$
(3) $\subset{\mathfrak{L}}_{1}^{\bar{0}}\oplus{\mathfrak{L}}_{1}^{\bar{1}},$

last equality being consequence of equation (1); and

$V_{{\mathfrak{C}}_{g}}:=\bigoplus\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}{\mathfrak{L}}_{g^{\prime}}=\bigoplus\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}({\mathfrak{L}}_{g^{\prime}}^{\bar{0}}\oplus{\mathfrak{L}}_{g^{\prime}}^{\bar{1}}).$

We denote by ${\mathfrak{L}}_{{\mathfrak{C}}_{g}}$ the following (graded)
subspace of $L$,

${\mathfrak{L}}_{{\mathfrak{C}}_{g}}:={\mathfrak{L}}_{{\mathfrak{C}}_{g},1}\oplus
V_{{\mathfrak{C}}_{g}}.$

######  Proposition 2.2.

Let $g\in\Sigma_{G}$. Then the following assertions hold.

  1. 1.

$[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{g}}]\subset{\mathfrak{L}}_{{\mathfrak{C}}_{g}}$.

  2. 2.

If $h\notin{\mathfrak{C}}_{g}$ then
$[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{h}}]=0$.

###### Proof.

1\. We have

$[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{g}}]=[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}\oplus
V_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}\oplus
V_{{\mathfrak{C}}_{g}}]\subset$ (4)
$[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}]+[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},V_{{\mathfrak{C}}_{g}}]+[V_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}]+[V_{{\mathfrak{C}}_{g}},V_{{\mathfrak{C}}_{g}}].$

Consider the above second summand
$[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},V_{{\mathfrak{C}}_{g}}]$. Taking into
account ${\mathfrak{L}}_{{\mathfrak{C}}_{g},1}\subset{\mathfrak{L}}_{1}$ and
$[{\mathfrak{L}}_{1},{\mathfrak{L}}_{g}]\subset{\mathfrak{L}}_{g}$ for any
$g\in\Sigma_{g}$, we have
$[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},V_{{\mathfrak{C}}_{g}}]\subset
V_{{\mathfrak{C}}_{g}}$. In a similar way
$[V_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}]\subset
V_{{\mathfrak{C}}_{g}}$ and so

$[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},\oplus
V_{{\mathfrak{C}}_{g}}]+[V_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}]\subset
V_{{\mathfrak{C}}_{g}}.$ (5)

Consider now the fourth summand
$[V_{{\mathfrak{C}}_{g}},V_{{\mathfrak{C}}_{g}}]$ in equation (4) and suppose
there exist $g^{\prime},g^{\prime\prime}\in{\mathfrak{C}}_{g}$ such that
$[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{g^{\prime\prime}}]\neq 0$. If
$g^{\prime\prime}=(g^{\prime})^{-1},$ clearly
$[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{g^{\prime\prime}}]=[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}]\subset{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}$.
Otherwise, if $g^{\prime\prime}\neq(g^{\prime})^{-1}$, then
$g^{\prime}g^{\prime\prime}\in\Sigma_{G}$ and Lemma 2.1 gives us
$g^{\prime}g^{\prime\prime}\in{\mathfrak{C}}_{g}$. Hence,
$[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{g^{\prime\prime}}]\subset{\mathfrak{L}}_{g^{\prime}g^{\prime\prime}}\subset
V_{{\mathfrak{C}}_{g}}$. In any case

$[V_{{\mathfrak{C}}_{g}},V_{{\mathfrak{C}}_{g}}]\subset{\mathfrak{L}}_{{\mathfrak{C}}_{g}}.$
(6)

Finally, let us consider the first summand
$[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}]$
in equation (4). We have

$[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}]=[\sum\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}],\sum\limits_{g^{\prime\prime}\in{\mathfrak{C}}_{g}}[{\mathfrak{L}}_{g^{\prime\prime}},{\mathfrak{L}}_{(g^{\prime\prime})^{-1}}]]\subset$
$\sum\limits_{{\tiny\begin{array}[]{c}g^{\prime},g^{\prime\prime}\in{\mathfrak{C}}_{g}\\\
\bar{i},\bar{j},\bar{k},\bar{l}\in{\hbox{\hueca Z}}_{2}\\\
\end{array}}}[[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{j}}],[{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{k}},{\mathfrak{L}}_{(g^{\prime\prime})^{-1}}^{\bar{l}}]],$

last equality being consequence of equation (3). Taking now into account super
Jacobi identity we get

$\sum\limits_{{\tiny\begin{array}[]{c}g^{\prime},g^{\prime\prime}\in{\mathfrak{C}}_{g}\\\
\bar{i},\bar{j},\bar{k},\bar{l}\in{\hbox{\hueca Z}}_{2}\\\
\end{array}}}[[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{j}}],[{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{k}},{\mathfrak{L}}_{(g^{\prime\prime})^{-1}}^{\bar{l}}]]\subset$
$\sum\limits_{{\tiny\begin{array}[]{c}g^{\prime},g^{\prime\prime}\in{\mathfrak{C}}_{g}\\\
\bar{i},\bar{j},\bar{k},\bar{l}\in{\hbox{\hueca Z}}_{2}\\\
\end{array}}}([{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},[{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{j}},[{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{k}},{\mathfrak{L}}_{(g^{\prime\prime})^{-1}}^{\bar{l}}]]]+[{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{j}},[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},[{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{k}},{\mathfrak{L}}_{(g^{\prime\prime})^{-1}}^{\bar{l}}]]])\subset$
$\sum\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}([{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{(g^{\prime})^{-1}}^{{\bar{j}}+{\bar{k}}+{\bar{l}}}]+[{\mathfrak{L}}_{(g^{\prime})^{-1}}^{{\bar{j}}},{\mathfrak{L}}_{g^{\prime}}^{{\bar{i}}+{\bar{k}}+{\bar{l}}}])\subset\sum\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}]={\mathfrak{L}}_{{\mathfrak{C}}_{g},1}.$

That is,

$[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}]\subset{\mathfrak{L}}_{{\mathfrak{C}}_{g},1}$
(7)

From equations (4)-(7) we conclude
$[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{g}}]\subset{\mathfrak{L}}_{{\mathfrak{C}}_{g}}$.

2\. We have as in 1. that

$[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{h}}]\subset[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},{\mathfrak{L}}_{{\mathfrak{C}}_{h},1}]+[{\mathfrak{L}}_{{\mathfrak{C}}_{g},1},V_{{\mathfrak{C}}_{h}}]+[V_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{{\mathfrak{C}}_{h},1}]+[V_{{\mathfrak{C}}_{g}},V_{{\mathfrak{C}}_{h}}].$
(8)

Let us suppose that there exist $g^{\prime}\in{\mathfrak{C}}_{g}$ and
$h^{\prime}\in{\mathfrak{C}}_{h}$ such that
$[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{h^{\prime}}]\neq 0$. Then
$g^{\prime}h^{\prime}\in\Sigma_{G}$ and we have as consequence of Lemma 2.1
that $g$ is connected to $h$, a contradiction. From here
$[V_{{\mathfrak{C}}_{g}},V_{{\mathfrak{C}}_{h}}]=0$. Taking into account this
equality and the fact $(g^{\prime})^{-1}\in{\mathfrak{C}}_{g}$ for any
$g^{\prime}\in{\mathfrak{C}}_{g}$, we can argue with super Jacobi identity in
$[[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}],{\mathfrak{L}}_{h^{\prime}}]$,
in a similar way to item 1., to get
$[[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}],{\mathfrak{L}}_{h^{\prime}}]=0$.
Now a same argument can be applied to verify
$[[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}],[{\mathfrak{L}}_{h^{\prime}},{\mathfrak{L}}_{(h^{\prime})^{-1}}]]=0.$
Taking into account equation (8) we have proved 2. ∎

Proposition 2.2-1 let us assert that for any $g\in\Sigma_{g}$,
${\mathfrak{L}}_{{\mathfrak{C}}_{g}}$ is a (graded) subalgebra of
${\mathfrak{L}}$ that we call the subalgebra of ${\mathfrak{L}}$ associated to
${\mathfrak{C}}_{g}$.

######  Theorem 2.1.

The following assertions hold.

  1. 1.

For any $g\in\Sigma_{G}$, the graded subalgebra
${\mathfrak{L}}_{{\mathfrak{C}}_{g}}={\mathfrak{L}}_{{\mathfrak{C}}_{g},1}\oplus
V_{{\mathfrak{C}}_{g}}$ of ${\mathfrak{L}}$ associated to ${\mathfrak{C}}_{g}$
is a graded ideal of ${\mathfrak{L}}$.

  2. 2.

If ${\mathfrak{L}}$ is gr-simple, then there exists a $\Sigma_{G}$-connection
from $g$ to $g^{\prime}$ for any $g,g^{\prime}\in\Sigma_{G}$, and
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$.

###### Proof.

1\. Taking into account Proposition 2.2 we have

$[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}]=[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{1}\oplus(\bigoplus\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}{\mathfrak{L}}_{g^{\prime}})\oplus(\bigoplus\limits_{h\notin{\mathfrak{C}}_{g}}{\mathfrak{L}}_{h})]=$
$=[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{1}]\oplus(\bigoplus\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{g^{\prime}}])\oplus(\bigoplus\limits_{h\notin{\mathfrak{C}}_{g}}[{\mathfrak{L}}_{{\mathfrak{C}}_{g}},{\mathfrak{L}}_{h}])\subset{\mathfrak{L}}_{{\mathfrak{C}}_{g}}.$

2\. The gr-simplicity of ${\mathfrak{L}}$ implies
${\mathfrak{L}}_{{\mathfrak{C}}_{g}}={\mathfrak{L}}$. From here
${\mathfrak{C}}_{g}=\Sigma_{G}$ and
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$.
∎

######  Theorem 2.2.

For the complementary vector space $\mathcal{U}$ of $span_{\hbox{\hueca
K}}\\{[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]:g\in\Sigma_{G}\\}$ in
${\mathfrak{L}}_{1}$, we have

${\mathfrak{L}}=\mathcal{U}+\sum\limits_{g\in\Sigma_{G}/\sim}I_{[g]},$

where any $I_{[g]}$ is one of the graded ideals
${\mathfrak{L}}_{{\mathfrak{C}}_{g}}$ of ${\mathfrak{L}}$ described in Theorem
2.1, satisfying $[I_{[g]},I_{[g^{\prime}]}]=0$ if $[g]\neq[g^{\prime}]$.

###### Proof.

By Proposition 2.1, we can consider the quotient set
$\Sigma_{G}/\sim:=\\{[g]:g\in\Sigma_{G}\\}$. Let us denote by
$I_{[g]}:={\mathfrak{L}}_{{\mathfrak{C}}_{g}}.$ We have $I_{[g]}$ is well
defined and, by Theorem 2.1-1, a graded ideal of ${\mathfrak{L}}$. Therefore

${\mathfrak{L}}=\mathcal{U}+\sum\limits_{[g]\in\Sigma_{G}/\sim}I_{[g]}.$

By applying Proposition 2.2-2 we also obtain $[I_{[g]},I_{[g^{\prime}]}]=0$ if
$[g]\neq[g^{\prime}]$. ∎

Let us denote by
${\mathcal{Z}}({\mathfrak{L}})=\\{v\in{\mathfrak{L}}:[v,{\mathfrak{L}}]=0\\}$
the center of ${\mathfrak{L}}$.

######  Corollary 2.1.

If ${\mathcal{Z}}({\mathfrak{L}})=0$ and
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$,
then ${\mathfrak{L}}$ is the direct sum of the graded ideals given in Theorem
2.1-1,

${\mathfrak{L}}=\bigoplus\limits_{[g]\in\Sigma_{G}/\sim}I_{[g]},$

which satisfy $[I_{[g]},I_{[g^{\prime}]}]=0$ if $[g]\neq[g^{\prime}]$.

###### Proof.

From
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$
it is clear that ${\mathfrak{L}}=\sum\limits_{[g]\in\Sigma_{G}/\sim}I_{[g]}$.
The direct character of the sum now follows from the facts
$[I_{[g]},I_{[g^{\prime}]}]=0$, if $[g]\neq[g^{\prime}]$, and
${\mathcal{Z}}({\mathfrak{L}})=0$. ∎

##  3 The gr-simple components

The study of the structure of this kind of algebras has been reduced to
consider those satisfying that the $G$-support has all of its elements
$\Sigma_{G}$-connected. It is a natural question if these algebras are gr-
simple. We study this problem in this section.

######  Lemma 3.1.

Let ${\mathfrak{L}}$ be a graded Lie superalgebra with
${\mathcal{Z}}({\mathfrak{L}})=0$ and
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$.
If $I$ is a graded ideal of ${\mathfrak{L}}$ such that
$I\subset{\mathfrak{L}}_{1}$, then $I=\\{0\\}.$

###### Proof.

Suppose there exists a nonzero graded ideal $I$ of ${\mathfrak{L}}$ such that
$I\subset{\mathfrak{L}}_{1}$. We have
$[I,\bigoplus\limits_{g\in\Sigma_{G}}{\mathfrak{L}}_{g}]\subset
I\subset{\mathfrak{L}}_{1}$, therefore
$[I,\bigoplus\limits_{g\in\Sigma_{G}}{\mathfrak{L}}_{g}]\subset{\mathfrak{L}}_{1}\cap(\bigoplus\limits_{g\in\Sigma_{G}}{\mathfrak{L}}_{g})=0$.
The fact ${\mathcal{Z}}({\mathfrak{L}})=0$ implies $[I,{\mathfrak{L}}_{1}]\neq
0$. Taking account
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$,
there exists $g_{0}\in\Sigma_{G}$ such that
$[I,[{\mathfrak{L}}_{g_{0}},{\mathfrak{L}}_{g_{0}^{-1}}]]\neq 0$. By writing
$I=I^{\bar{0}}\oplus I^{\bar{1}}$ with
$I^{\bar{i}}=I\cap{\mathfrak{L}}^{\bar{i}}$, $i\in{\hbox{\hueca Z}}_{2}$, and
taking into account equation (1) we have
$[I^{\bar{i}},[{\mathfrak{L}}_{g_{0}}^{\bar{j}},{\mathfrak{L}}_{g_{0}^{-1}}^{\bar{k}}]]\neq
0$ for some ${\bar{i}},{\bar{j}},{\bar{k}}\in{\hbox{\hueca Z}}_{2}$. Super
Jacobi identity gives us now that either
$0\neq[I^{\bar{i}},{\mathfrak{L}}_{g_{0}}^{\bar{j}}]\subset{\mathfrak{L}}_{g_{0}}\cap{\mathfrak{L}}_{1}$
or
$0\neq[I^{\bar{i}},{\mathfrak{L}}_{g_{0}^{-1}}^{\bar{k}}]\subset{\mathfrak{L}}_{g_{0}^{-1}}\cap{\mathfrak{L}}_{1}$,
a contradiction. Therefore $I=\\{0\\}.$ ∎

Let us introduce the concepts of $\Sigma_{G}$-multiplicativity and maximal
length in the framework of graded Lie superalgebras, in a similar way to the
ones for graded Lie algebras [6], split Lie superalgebras [9], and split Lie
triple systems [5] among other contexts. Recall that we denote by
$\Sigma_{G}^{\bar{0}}:=\\{g\in\Sigma_{G}:{\mathfrak{L}}_{g}^{\bar{0}}\neq
0\\}$ and by
$\Sigma_{G}^{\bar{1}}:=\\{g\in\Sigma_{G}:{\mathfrak{L}}_{g}^{\bar{1}}\neq
0\\}$.

######  Definition 3.1.

We say that a graded Lie superalgebra ${\mathfrak{L}}$ is of maximal length if
$\dim{\mathfrak{L}}_{g}^{\bar{i}}\in\\{0,1\\}$ for any $g\in\Sigma_{G}$ and
${\bar{i}}\in{\hbox{\hueca Z}}_{2}$.

Observe that, for a graded Lie superalgebra of maximal length
${\mathfrak{L}}$, the symmetry of $\Sigma_{G}$ gives us that given some
$g\in\Sigma_{G}$,

either ${\rm dim}({\mathfrak{L}}_{g})={\rm dim}({\mathfrak{L}}_{g^{-1}})=1$ in
the case $g\notin\Sigma_{G}^{\bar{0}}\cap\Sigma_{G}^{\bar{1}}$, or (9) ${\rm
dim}({\mathfrak{L}}_{g})={\rm dim}({\mathfrak{L}}_{g^{-1}})=2$ in the case
$g\in\Sigma_{G}^{\bar{0}}\cap\Sigma_{G}^{\bar{1}}$.

Also observe that equation (2) let us assert that any nonzero graded ideal $I$
of ${\mathfrak{L}}$ is of the form

$I=(I_{1}^{\bar{0}}\oplus(\bigoplus\limits_{g\in\Sigma_{I}^{\bar{0}}}{\mathfrak{L}}_{g}^{\bar{0}}))\oplus(I_{1}^{\bar{1}}\oplus(\bigoplus\limits_{g^{\prime}\in\Sigma_{I}^{\bar{1}}}{\mathfrak{L}}_{g^{\prime}}^{\bar{1}}))$
(10)

${\rm where}\hskip
2.84544pt\Sigma_{I}^{\bar{i}}=\\{h\in\Sigma_{G}^{\bar{i}}:I\cap{\mathfrak{L}}_{h}^{\bar{i}}\neq
0\\},{\bar{i}}\in{\hbox{\hueca Z}}_{2}$.

######  Definition 3.2.

We say that a graded Lie superalgebra ${\mathfrak{L}}$ is
$\Sigma_{G}$-multiplicative if given $g\in\Sigma_{G}^{\bar{i}}$ and
$g^{\prime}\in\Sigma_{G}^{\bar{j}},{\bar{i}},{\bar{j}}\in{\hbox{\hueca
Z}}_{2}$, such that $gg^{\prime}\in\Sigma_{G}$, then
$[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{\prime}}^{\bar{j}}]\neq 0$.

As examples of $\Sigma_{G}$-multiplicative graded Lie superalgebras with
maximal length we have the split Lie superalgebras considered in [9], the
graded Lie algebras in [6], the semisimple separable $L^{*}$-algebras [23] and
the semisimple locally finite split Lie algebras over a field of
characteristic zero [24].

######  Lemma 3.2.

Let ${\mathfrak{L}}$ be $\Sigma_{G}$-multiplicative and of maximal length. If
$\Sigma_{G}$ has all of its elements $\Sigma_{G}$-connected, then either ${\rm
dim}({\mathfrak{L}}_{g})=1$ for any $g\in\Sigma_{G}$ or ${\rm
dim}({\mathfrak{L}}_{g})=2$ for any $g\in\Sigma_{G}$.

###### Proof.

Suppose there exists $g\in\Sigma_{G}$ such that ${\rm
dim}({\mathfrak{L}}_{g})=2$. Hence, ${\rm dim}({\mathfrak{L}}_{g^{-1}})=2$ and
we can write
${\mathfrak{L}}_{g}={\mathfrak{L}}_{g}^{\bar{0}}\oplus{\mathfrak{L}}_{g}^{\bar{1}}$
with any ${\mathfrak{L}}_{g}^{\bar{i}}\neq 0$. Given now any
$g^{\prime}\in\Sigma_{G}\setminus\\{g,g^{-1}\\}$, the fact that $g$ and
$g^{\prime}$ are $\Sigma_{G}$-connected gives us a $\Sigma_{G}$-connection
$\\{g_{1},g_{2},....,g_{r}\\}$ from $g$ to $g^{\prime}$ such that

$g_{1}=g,$ $g_{1}g_{2},g_{1}g_{2}g_{3},...,g_{1}g_{2}g_{3}\cdots
g_{r-1}\in\Sigma_{G}$

and

$g_{1}g_{2}g_{3}\cdots g_{r}\in\\{g^{\prime},(g^{\prime})^{-1}\\}.$

Consider $g_{1},g_{2}$ and $g_{1}g_{2}$. Since $g_{2}\in\Sigma_{G}$, some
${\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}}\neq 0$ with
${\bar{i}_{2}}\in{\hbox{\hueca Z}}_{2}$, and so
$g_{2}\in\Sigma_{G}^{\bar{i}_{2}}$. We have $g_{1}=g\in\Sigma_{G}^{\bar{0}}$
and $g_{2}\in\Sigma_{G}^{\bar{i}_{2}}$ such that $g_{1}g_{2}\in\Sigma_{G}$.
Then the $\Sigma_{G}$-multiplicativity of ${\mathfrak{L}}$ gives us

$0\neq[{\mathfrak{L}}_{g_{1}}^{\bar{0}},{\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}}]\subset{\mathfrak{L}}_{g_{1}g_{2}}^{\bar{i}_{2}}$
(11)

Hence, the maximal length of ${\mathfrak{L}}$ shows
$\dim{\mathfrak{L}}_{g_{1}g_{2}}^{\bar{i}_{2}}=1$ and so
$0\neq[{\mathfrak{L}}_{g_{1}}^{\bar{0}},{\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}}]={\mathfrak{L}}_{g_{1}g_{2}}^{\bar{i}_{2}}.$
We can argue in a similar way from $g_{1}g_{2},g_{3}$ and $g_{1}g_{2}g_{3}$ to
get

$0\neq[[{\mathfrak{L}}_{g_{1}}^{\bar{0}},{\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}}],{\mathfrak{L}}_{g_{2}}^{\bar{i}_{3}}]={\mathfrak{L}}_{g_{1}g_{2}g_{3}}^{\bar{i}_{2}+\bar{i}_{3}}$

for some ${\bar{i}_{3}}\in{\hbox{\hueca Z}}_{2}$. Following this process with
the $\Sigma_{G}$-connection $\\{g_{1},....,g_{r}\\}$ we obtain that

$0\neq[[\cdots[[{\mathfrak{L}}_{g_{1}}^{\bar{0}},{\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}}],{\mathfrak{L}}_{g_{2}}^{\bar{i}_{3}}],\cdots],{\mathfrak{L}}_{g_{1}}^{\bar{i}_{r}}]={\mathfrak{L}}_{g_{1}g_{2}g_{3}\cdots
g_{r}}^{\bar{i}_{2}+\cdots+\bar{i}_{r}}$

and so either
$0\neq{\mathfrak{L}}_{g^{\prime}}^{\bar{i}_{2}+\cdots+\bar{i}_{r}}$ or
$0\neq{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{i}_{2}+\cdots+\bar{i}_{r}}.$
That is, for any $g^{\prime}\in\Sigma_{G}\setminus\\{g,g^{-1}\\}$ we have that

$0\neq{\mathfrak{L}}_{\xi}^{\bar{i}_{2}+\cdots+\bar{i}_{r}}$ for some
$\xi\in\\{g^{\prime},(g^{\prime})^{-1}\\}$. (12)

If we argue with the $\Sigma_{G}$-connection $\\{g_{1},g_{2},....,g_{r}\\}$ as
above, but starting in equation (11) with ${\mathfrak{L}}_{g_{1}}^{\bar{1}}$
instead of ${\mathfrak{L}}_{g_{1}}^{\bar{0}}$ we get
$0\neq{\mathfrak{L}}_{\xi}^{\bar{1}+\bar{i}_{2}+\cdots+\bar{i}_{r}}$. Hence,
and taking into account equation (12) we have ${\rm
dim}({\mathfrak{L}}_{\xi})=2$ and, by the symmetry of $\Sigma_{G}$, ${\rm
dim}({\mathfrak{L}}_{\xi^{-1}})=2$. We conclude ${\rm
dim}({\mathfrak{L}}_{g^{\prime}})=2$ and so ${\rm dim}({\mathfrak{L}}_{g})=2$
for any $g\in\Sigma_{G}$. Equation (9) completes the proof. ∎

Let ${\mathfrak{L}}$ be $\Sigma_{G}$-multiplicative, of maximal length, with
${\mathcal{Z}}({\mathfrak{L}})=0$, satisfying
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$
and with all of the elements in its $G$-support $\Sigma_{G}$-connected, and
consider a nonzero graded ideal $I$ of ${\mathfrak{L}}$. By equation (10) we
can write
$I=(I_{1}^{\bar{0}}\oplus(\bigoplus\limits_{g\in\Sigma_{I}^{\bar{0}}}{\mathfrak{L}}_{g}^{\bar{0}}))\oplus(I_{1}^{\bar{1}}\oplus(\bigoplus\limits_{g^{\prime}\in\Sigma_{I}^{\bar{1}}}{\mathfrak{L}}_{g^{\prime}}^{\bar{1}}))$
${\rm where}\hskip
2.84544pt\Sigma_{I}^{\bar{i}}=\\{h\in\Sigma_{G}^{\bar{i}}:I\cap{\mathfrak{L}}_{h}^{\bar{i}}\neq
0\\},{\bar{i}}\in{\hbox{\hueca Z}}_{2}$. Furthermore, Lemma 3.1 let us assert
that $\Sigma_{I}^{\bar{i}}\neq\emptyset$ for some ${\bar{i}}\in{\hbox{\hueca
Z}}_{2}$. So, we can take $g_{0}\in\Sigma_{I}^{\bar{i}}$ such that

$0\neq{\mathfrak{L}}_{g_{0}}^{\bar{i}}\subset I.$ (13)

For any $g^{\prime}\in\Sigma_{G}\setminus\\{g_{0},g_{0}^{-1}\\}$, the fact
that $g_{0}$ and $g^{\prime}$ are $\Sigma_{G}$-connected gives us a
$\Sigma_{G}$-connection

$\\{g_{1},g_{2},....,g_{r}\\}$ (14)

from $g_{0}$ to $g^{\prime}$ such that

$g_{1}=g_{0},$ $g_{1}g_{2},g_{1}g_{2}g_{3},...,g_{1}g_{2}g_{3}\cdots
g_{r-1}\in\Sigma_{G}$

and

$g_{1}g_{2}g_{3}\cdots g_{r}\in\\{g^{\prime},(g^{\prime})^{-1}\\}.$

Consider $g_{1},g_{2}$ and $g_{1}g_{2}$. By arguing with the
$\Sigma_{G}$-multiplicativity and maximal length of ${\mathfrak{L}}$ as in the
proof of Lemma 3.2, we get
$0\neq[{\mathfrak{L}}_{g_{1}}^{\bar{i}},{\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}}]={\mathfrak{L}}_{g_{1}g_{2}}^{\bar{i}+\bar{i}_{2}}$
From here, equation (13) let us conclude

$0\neq{\mathfrak{L}}_{g_{1}g_{2}}^{\bar{i}+\bar{i}_{2}}\subset I.$

We can argue in a similar way from $g_{1}g_{2},g_{3}$ and $g_{1}g_{2}g_{3}$ to
get

$0\neq{\mathfrak{L}}_{g_{1}g_{2}g_{3}}^{\bar{i}+\bar{i}_{2}+\bar{i}_{3}}\subset
I$

for some ${\bar{i}_{3}}\in{\hbox{\hueca Z}}_{2}$. Following this process with
the $\Sigma_{G}$-connection $\\{g_{1},....,g_{r}\\}$ we obtain that

$0\neq{\mathfrak{L}}_{g_{1}g_{2}g_{3}\cdots g_{r}}^{\bar{p}}\subset I,$ (15)

$\bar{p}=\bar{i}+\bar{i}_{2}+\bar{i}_{3}+\cdots+\bar{i}_{r}$, and so either
$0\neq{\mathfrak{L}}_{g^{\prime}}^{\bar{p}}\subset I$ or
$0\neq{\mathfrak{L}}_{(g^{\prime})^{-1}}^{\bar{p}}\subset I$ for some
${\bar{p}}\in{\hbox{\hueca Z}}_{2}$. That is, for any
$g^{\prime}\in\Sigma_{G}$ we have that

$0\neq{\mathfrak{L}}_{\xi}^{\bar{p}}\subset I$ for some
$\xi\in\\{g^{\prime},(g^{\prime})^{-1}\\}$ and some ${\bar{p}}\in{\hbox{\hueca
Z}}_{2}$. (16)

Taking into account Lemma 3.2 we can distinguish two possibilities.

In the first one ${\rm dim}({\mathfrak{L}}_{g})=1$ for any $g\in\Sigma_{G}$
and so equation (16) gives us that, in this first possibility,

either ${\mathfrak{L}}_{g}\subset I$ or ${\mathfrak{L}}_{g^{-1}}\subset I$ for
any $g\in\Sigma_{G}$. (17)

In the second possibility, ${\rm dim}({\mathfrak{L}}_{g})=2$ for any
$g\in\Sigma_{G}$. Given now any
$g^{\prime}\in\Sigma_{G}\setminus\\{g_{0},g_{0}^{-1}\\}$ and the
$\Sigma_{G}$-connection (14) from $g_{0}$ to $g^{\prime}$, the
$\Sigma_{G}$-multiplicativity of ${\mathfrak{L}}$ let us also get in a first
step
$0\neq[{\mathfrak{L}}_{g_{1}}^{\bar{i}},{\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}+\bar{1}}]={\mathfrak{L}}_{g_{1}g_{2}}^{\bar{i}+\bar{i}_{2}+\bar{1}}\subset
I$, in a second step
$0\neq[{\mathfrak{L}}_{g_{1}g_{2}}^{\bar{i}+\bar{i}_{2}+\bar{1}},{\mathfrak{L}}_{g_{3}}^{\bar{i}_{3}}]={\mathfrak{L}}_{g_{1}g_{2}g_{3}}^{\bar{i}+\bar{i}_{2}+\bar{i}_{3}+\bar{1}}\subset
I$, and finally $0\neq{\mathfrak{L}}_{g_{1}g_{2}g_{3}\cdots
g_{r}}^{\bar{p}+\bar{1}}\subset I.$ Taking into account equation (15), we have
showed that in this second possibility

either ${\mathfrak{L}}_{g^{\prime}}\subset I$ or
${\mathfrak{L}}_{(g^{\prime})^{-1}}\subset I$ for any
$g^{\prime}\in\Sigma_{G}\setminus\\{g_{0},g_{0}^{-1}\\}$. (18)

Observe that it remains to study if equation (18) also holds for $g_{0}$. To
do that, let us suppose the cardinal of $\Sigma_{G}$, denoted by
$|\Sigma_{G}|$, is greater than 2. Then there exists some
${g^{\prime}}\in\Sigma_{G}\setminus\\{g_{0},g_{0}^{-1}\\}$ and, by the above,
a $\Sigma_{G}$-connection $\\{g_{0},g_{2},....,g_{r}\\}$ from $g_{0}$ to
$g^{\prime}$ such that
${\mathfrak{L}}_{g^{\prime}}^{\bar{0}}\oplus{\mathfrak{L}}_{g^{\prime}}^{\bar{1}}\subset
I$ with any ${\mathfrak{L}}_{g^{\prime}}^{\bar{i}}\neq 0$ being also
$0\neq[[\cdots[[{\mathfrak{L}}_{g_{0}}^{\bar{i}},{\mathfrak{L}}_{g_{2}}^{\bar{i}_{2}}],{\mathfrak{L}}_{g_{3}}^{\bar{i}_{3}}],\cdots],{\mathfrak{L}}_{g_{r}}^{\bar{i}_{r}}]={\mathfrak{L}}_{g^{\prime}}^{\bar{p}}$,
$\bar{p}=\bar{i}+\bar{i}_{2}\cdots+\bar{i}_{r}$. From here, we also have the
$\Sigma_{G}$-connection $\\{g_{0}g_{2}\cdot\cdot\cdot
g_{r},g_{r}^{-1},g_{r-1}^{-1},...,g_{2}^{-1}\\}\subset\Sigma_{G}$ which
satisfies $g_{0}g_{2}\cdot\cdot\cdot g_{r},g_{0}g_{2}\cdot\cdot\cdot
g_{r-1},...,g_{0}\in\Sigma_{G}$ and $g_{0}g_{2}\cdot\cdot\cdot
g_{r}=g^{\prime}$. By $\Sigma_{G}$-multiplicativity, and taking into account
${\mathfrak{L}}_{{g^{\prime}}}^{\bar{0}}\oplus{\mathfrak{L}}_{{g^{\prime}}}^{\bar{1}}\subset
I$ with any ${\mathfrak{L}}_{{g^{\prime}}}^{\bar{i}}\neq 0$,
$0\neq[[...[[{\mathfrak{L}}_{g_{0}g_{2}\cdot\cdot\cdot
g_{r}}^{\bar{p}+\bar{1}},{\mathfrak{L}}_{g_{r}^{-1}}^{\bar{i}_{r}}],{\mathfrak{L}}_{g_{r-1}^{-1}}^{\bar{i}_{r-1}}],...],{\mathfrak{L}}_{g_{2}^{-1}}^{\bar{i}_{2}}]={\mathfrak{L}}_{g_{0}}^{\bar{i}+\bar{1}}\subset
I.$ From here ${\mathfrak{L}}_{g_{0}}\subset I$. Let us observe that the above
argument also shows that, (under the assumption $|\Sigma_{G}|>2$),

in case ${\mathfrak{L}}_{g_{0}}^{\bar{i}}\subset I$ for some
$g_{0}\in\Sigma_{G}$ and some ${\bar{i}}\in{\hbox{\hueca Z}}_{2}$, then
${\mathfrak{L}}_{g_{0}}\subset I$. (19)

Summarizing the above paragraphs, equations (17), (18) and (19) let us assert
that in case $|\Sigma_{G}|>2$ then

either ${\mathfrak{L}}_{g}\subset I$ or ${\mathfrak{L}}_{g^{-1}}\subset I$ for
any $g\in\Sigma_{G}$. (20)

From now on we are also going to suppose $|\Sigma_{G}|>2$, (the easy case in
which $|\Sigma_{G}|<2$ will be consider below in Lemma 3.6). Then equation
(20) let us denote by

$\Sigma_{I}:=\\{g\in\Sigma_{G}:{\mathfrak{L}}_{g}\subset I\\}$

and assert, taking into account the fact
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$,
that

${\mathfrak{L}}_{1}\subset I.$ (21)

Let us also denote by

$J:=\bigoplus\limits_{g^{\prime}\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g^{\prime}}.$

Observe that equation (19), joint with the maximal lenght of ${\mathfrak{L}}$
and the graded character of ${\mathfrak{L}}_{g^{\prime}}$ and of $I$, let us
assert that if $g^{\prime}\in\Sigma_{G}\setminus\Sigma_{I}$ then
${\mathfrak{L}}_{g^{\prime}}\cap I=\\{0\\}$. From here, taking also into
account equation (21), we can write

$I={\mathfrak{L}}_{1}\oplus(\bigoplus\limits_{g\in\Sigma_{I}}{\mathfrak{L}}_{g}).$
(22)

######  Lemma 3.3.

The following assertions hold.

  * (i)

For any $g^{\prime}\in\Sigma_{G}\setminus\Sigma_{I}$ and
${\bar{i}}\in{\hbox{\hueca Z}}_{2}$, we have
$[{\mathfrak{L}}_{1},{\mathfrak{L}}_{g^{\prime}}^{\bar{i}}]=0$.

  * (ii)

For any $g^{\prime},g^{\prime\prime}\in\Sigma_{G}\setminus\Sigma_{I}$ and
${\bar{i}},{\bar{j}}\in{\hbox{\hueca Z}}_{2}$, we have
$[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{j}}]\subset
J$.

  * (iii)

For any $g\in\Sigma_{I},g^{\prime}\in\Sigma_{G}\setminus\Sigma_{I}$ with
$g^{\prime}\neq g^{-1}$ and ${\bar{i}},{\bar{j}}\in{\hbox{\hueca Z}}_{2}$, we
have $[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{\prime}}^{\bar{j}}]=0$.

###### Proof.

(i) Suppose $[{\mathfrak{L}}_{1},{\mathfrak{L}}_{g^{\prime}}^{\bar{i}}]\neq
0$, then
$0\neq[{\mathfrak{L}}_{1}^{\bar{k}},{\mathfrak{L}}_{g^{\prime}}^{\bar{i}}]={\mathfrak{L}}_{g^{\prime}}^{\bar{k}+\bar{i}}$
for some ${\bar{k}}\in{\hbox{\hueca Z}}_{2}$. Equation (21) gives us
${\mathfrak{L}}_{g^{\prime}}^{\bar{k}+\bar{i}}\subset I$ and so, taking into
account equation (19), $g^{\prime}\in\Sigma_{I}$, a contradiction. Hence
$[{\mathfrak{L}}_{1},{\mathfrak{L}}_{g^{\prime}}^{\bar{i}}]=0$.

(ii) If
$[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{j}}]=0$
then
$[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{j}}]\subset
J$. So, let consider the case
$[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{j}}]\neq
0$, being then

$0\neq[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{j}}]={\mathfrak{L}}_{g^{\prime}g^{\prime\prime}}^{\bar{i}+\bar{j}},$

and suppose $g^{\prime}g^{\prime\prime}\in\Sigma_{I}$. By
$\Sigma_{G}$-multiplicativity,
$[{\mathfrak{L}}_{g^{\prime}g^{\prime\prime}}^{\bar{i}+\bar{j}},{\mathfrak{L}}_{(g^{\prime\prime})^{-1}}^{\bar{j}}]={\mathfrak{L}}_{g^{\prime}}^{\bar{i}}\subset
I$. From here, $g^{\prime}\in\Sigma_{I}$, a contradiction. We conclude
$g^{\prime}g^{\prime\prime}\in\Sigma_{G}\setminus\Sigma_{I}$ and so
$[{\mathfrak{L}}_{g^{\prime}}^{\bar{i}},{\mathfrak{L}}_{g^{\prime\prime}}^{\bar{j}}]\subset
J.$

(iii) If
$[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{\prime}}^{\bar{j}}]\neq 0$,
we have as in (ii) that $g^{\prime}\in\Sigma_{I}$ which is a contradiction.
Hence,
$[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{\prime}}^{\bar{j}}]=0$. ∎

######  Lemma 3.4.

If $g\in\Sigma_{I}$ and $g^{-1}\in\Sigma_{G}\setminus\Sigma_{I}$ then
$[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]=0$ for any
$\bar{i},\bar{j}\in{\hbox{\hueca Z}}_{2}$.

###### Proof.

Let us consider $g\in\Sigma_{I}$ and $g^{-1}\in\Sigma_{G}\setminus\Sigma_{I}$.
Lemma 3.3-(i) gives us
$[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{h}^{\bar{k}}]=0$
in case $h\in\Sigma_{G}\setminus\Sigma_{I}$ for any
$\bar{i},\bar{j},\bar{k}\in{\hbox{\hueca Z}}_{2}$. Consider now any
$f\in\Sigma_{I}$ such that $f\neq g$ and any $\bar{s}\in{\hbox{\hueca
Z}}_{2}$, then we have

$[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{f}^{\bar{s}}]\subset[[{\mathfrak{L}}_{g^{-1}}^{\bar{j}},{\mathfrak{L}}_{f}^{\bar{s}}],{\mathfrak{L}}_{g}^{\bar{i}}]+[[{\mathfrak{L}}_{f}^{\bar{s}},{\mathfrak{L}}_{g}^{\bar{i}}],{\mathfrak{L}}_{g^{-1}}^{\bar{j}}].$

Taking into account that $f\in\Sigma_{I}$ and that in case
$[{\mathfrak{L}}_{f}^{\bar{s}},{\mathfrak{L}}_{g}^{\bar{i}}]\neq 0$ then
$fg\in\Sigma_{I}$, Lemma 3.3-(iii) shows that both above summands are zero and
so
$[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{f}^{\bar{s}}]=0$.
That is,

$[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{h}^{\bar{k}}]=0$
for any $h\in\Sigma_{G}\setminus\\{g\\}$ and $\bar{k}\in{\hbox{\hueca
Z}}_{2}$. (23)

We also have for any $l\in\Sigma_{G}$ and $\bar{s}\in{\hbox{\hueca Z}}_{2}$
that

$[[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{g}^{\bar{k}}],{\mathfrak{L}}_{l}^{\bar{s}}]\subset[[{\mathfrak{L}}_{g}^{\bar{k}},{\mathfrak{L}}_{l}^{\bar{s}}],[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]]+[[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{l}^{\bar{s}}],{\mathfrak{L}}_{g}^{\bar{k}}].$
(24)

Consider the second summand in equation (24). From equation (23),

$[[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{l}^{\bar{s}}],{\mathfrak{L}}_{g}^{\bar{k}}]=0$

up to maybe for $l=g$. In this case, if

$0\neq[[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{g}^{\bar{s}}],{\mathfrak{L}}_{g}^{\bar{k}}]\subset{\mathfrak{L}}_{g^{2}}^{\bar{i}+\bar{j}+\bar{s}+\bar{k}},$

being then ${\mathfrak{L}}_{g^{2}}^{\bar{i}+\bar{j}+\bar{s}+\bar{k}}\subset
I$. We have by $\Sigma_{G}$-multiplicativity
$0\neq[{\mathfrak{L}}_{g^{2}}^{\bar{i}+\bar{j}+\bar{s}+\bar{k}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]$,
but Lemma 3.3-(iii) shows
$[{\mathfrak{L}}_{g^{2}}^{\bar{i}+\bar{j}+\bar{s}+\bar{k}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]=0$,
a contradiction. Hence

$[[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{g}^{\bar{s}}],{\mathfrak{L}}_{g}^{\bar{k}}]=0,$

and so

$[[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{l}^{\bar{s}}],{\mathfrak{L}}_{g}^{\bar{k}}]=0$
for any $l\in\Sigma_{G}$. (25)

Consider now the first summand in equation (24). We have

$[[{\mathfrak{L}}_{g}^{\bar{k}},{\mathfrak{L}}_{l}^{\bar{s}}],[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]]\subset[{\mathfrak{L}}_{gl}^{\bar{k}+\bar{s}},[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]].$

But in case $l\neq g^{-1}$ we can assert
$[{\mathfrak{L}}_{gl}^{\bar{k}+\bar{s}},[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]]=0.$
Indeed, in the opposite case, that is,

$[{\mathfrak{L}}_{gl}^{\bar{k}+\bar{s}},[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]]\neq
0,$ (26)

we have that $gl\neq 1$ with
$0\neq{\mathfrak{L}}_{gl}^{\bar{k}+\bar{s}}\subset I$, and by super Jacobi
identity and Lemma 3.3-(iii) that
$[[{\mathfrak{L}}_{gl}^{\bar{k}+\bar{s}},{\mathfrak{L}}_{g}^{\bar{i}}],{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]\neq
0$, which implies

$[{\mathfrak{L}}_{g^{2}l}^{\bar{k}+\bar{s}+\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]\neq
0.$ (27)

Since by equation (26) and Lemma 3.3-(i), $l\neq g^{-2}$ and so $g^{2}l\neq
1$, we have ${g^{2}l}\in\Sigma_{I}$ and
$g^{-1}\in\Sigma_{G}\setminus\Sigma_{I}$. Then by Lemma 3.3-(iii) we get
$[{\mathfrak{L}}_{g^{2}l}^{\bar{k}+\bar{s}+\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]=0$,
which contradicts equation (27). So
$[{\mathfrak{L}}_{gl}^{\bar{k}+\bar{s}},[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]]=0$
and hence

$[[{\mathfrak{L}}_{g}^{\bar{k}},{\mathfrak{L}}_{l}^{\bar{s}}],[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]]=0$
if $l\neq g^{-1}$. (28)

Taking into account equations (24), (25) and (28) we have showed that

$[[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{g}^{\bar{k}}],{\mathfrak{L}}_{l}^{\bar{s}}]=0$
(29)

for any $l\in\Sigma_{G}\setminus\\{g^{-1}\\}$ and
$\bar{i},\bar{j},\bar{k},\bar{s}\in{\hbox{\hueca Z}}_{2}.$

Suppose now
$[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]\neq 0$ for
some $\bar{i},\bar{j}\in{\hbox{\hueca Z}}_{2}$. Since
${\mathcal{Z}}({\mathfrak{L}})=0$ and taking into account
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$
and super Jacobi identity, there exist $h\in\Sigma_{G}$ and
$\bar{k}\in{\hbox{\hueca Z}}_{2}$ such that
$[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{h}^{\bar{k}}]\neq
0.$ By equation (23) necessarily $h=g$ and so

$0\neq[[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}],{\mathfrak{L}}_{g}^{\bar{k}}]={\mathfrak{L}}_{g}^{\bar{i}+\bar{j}+\bar{k}}.$
(30)

Hence, equation (29) gives us
$[{\mathfrak{L}}_{g}^{\bar{i}+\bar{j}+\bar{k}},{\mathfrak{L}}_{l}^{\bar{s}}]=0$
for any $l\in\Sigma_{G}\setminus\\{g^{-1}\\}$ and $\bar{s}\in{\hbox{\hueca
Z}}_{2}.$ That is, we have showed that

in case $[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]\neq
0$ then $[{\mathfrak{L}}_{g}^{\bar{t}},{\mathfrak{L}}_{l}^{\bar{s}}]=0$ (31)

with ${\mathfrak{L}}_{g}^{\bar{t}}\neq 0,\bar{t}\in{\hbox{\hueca Z}}_{2}$, and
for any $l\in\Sigma_{G}\setminus\\{g^{-1}\\}$ and any $\bar{s}\in{\hbox{\hueca
Z}}_{2}$. However, since the cardinal of $\Sigma_{G}$ is greater than 2, there
exists $h\in\Sigma_{G}\setminus\\{g,g^{-1}\\}$. From here, we can take

$\\{g_{1},g_{2},...,g_{r}\\}\subset\Sigma_{G}$

a $\Sigma_{G}$-connection from $g$ to $h$, (which in particular implies
$g_{2}\neq g_{1}^{-1}$), satisfying $r\geq 2$. Then we have
$g_{1}=g\in\Sigma_{G}^{\bar{t}},g_{2}\in\Sigma_{G}^{\bar{p}}$ for some
$\bar{p}\in{\hbox{\hueca Z}}_{2}$, and $g_{1}g_{2}\in\Sigma_{G}$. Hence, the
$\Sigma_{G}$-multiplicativity of ${\mathfrak{L}}$ gives us

$[{\mathfrak{L}}_{g}^{\bar{t}},{\mathfrak{L}}_{g_{2}}^{\bar{p}}]\neq 0$

with $g_{2}\neq g^{-1}$, a contradiction with equation (31). Hence
$[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g^{-1}}^{\bar{j}}]=0$ for any
$\bar{i},\bar{j}\in{\hbox{\hueca Z}}_{2}$ and the proof is complete. ∎

######  Lemma 3.5.

If $g,g^{-1}\in\Sigma_{I}$ for some $g\in\Sigma_{G}$ then $I={\mathfrak{L}}$.

###### Proof.

Given any $g^{\prime}\in\Sigma_{G}\setminus\\{g,g^{-1}\\}$ there exists a
$\Sigma_{G}$-connection $\\{g_{1},...,g_{n}\\}$ with $g_{1}=g^{\prime}$ and
$g_{1}\cdots g_{n}\in\\{g,g^{-1}\\}$. From here, we can consider the
$\Sigma_{G}$-connection $\\{g_{1}\cdots g_{n},g_{n}^{-1},...,g_{2}^{-1}\\}$
from $g_{1}\cdots g_{n}$ to $g$. By Lemma 3.2 we can distinguish two
possibilities. In the first one ${\rm dim}({\mathfrak{L}}_{h})=1$ for any
$h\in\Sigma_{G}$ and so ${\mathfrak{L}}_{g_{1}\cdots
g_{n}}={\mathfrak{L}}_{g_{1}\cdots g_{n}}^{\bar{i}}$ for some
$\bar{i}\in{\hbox{\hueca Z}}_{2}$. The $\Sigma_{G}$-multiplicativity of
${\mathfrak{L}}$ let us assert

$[[...[[{\mathfrak{L}}_{g_{1}\cdots
g_{n}}^{\bar{i}},{\mathfrak{L}}_{g_{n}^{-1}}^{\bar{i}_{1}}],{\mathfrak{L}}_{g_{n-1}^{-1}}^{\bar{i}_{2}}],...],{\mathfrak{L}}^{\bar{i}_{n}}_{{g_{2}}^{-1}}]={\mathfrak{L}}_{g^{\prime}}^{\bar{j}}={\mathfrak{L}}_{g^{\prime}},$

for some $\bar{i}_{1},\bar{i}_{2},...,\bar{i}_{n},\bar{j}\in{\hbox{\hueca
Z}}_{2}$. Since ${\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}\subset I$ and so
${\mathfrak{L}}_{g_{1}\cdots g_{n}}^{\bar{i}}\subset I$ we get
${\mathfrak{L}}_{g^{\prime}}\subset I$. This fact joint with equation (21) let
us conclude $I={\mathfrak{L}}$. In the second possibility, ${\rm
dim}({\mathfrak{L}}_{h})=2$ for any $h\in\Sigma_{G}$. The
$\Sigma_{G}$-multiplicativity of ${\mathfrak{L}}$ let us now assert

$[[...[[{\mathfrak{L}}_{g_{1}\cdots
g_{n}}^{\bar{0}},{\mathfrak{L}}_{g_{n}^{-1}}^{\bar{i}_{1}}],{\mathfrak{L}}_{g_{n-1}^{-1}}^{\bar{i}_{2}}],...],{\mathfrak{L}}^{\bar{i}_{n}}_{{g_{2}}^{-1}}]={\mathfrak{L}}_{g^{\prime}}^{\bar{j}},$

for some $\bar{i}_{1},\bar{i}_{2},...,\bar{i}_{n},\bar{j}\in{\hbox{\hueca
Z}}_{2}$. As above, the fact
${\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}\subset I$ implies
${\mathfrak{L}}_{g_{1}\cdots g_{n}}^{\bar{0}}\subset I$ and then
${\mathfrak{L}}_{g^{\prime}}^{\bar{j}}\subset I$. In a similar way we have

$[[...[[{\mathfrak{L}}_{g_{1}\cdots
g_{n}}^{\bar{1}},{\mathfrak{L}}_{g_{n}^{-1}}^{\bar{i}_{1}}],{\mathfrak{L}}_{g_{n-1}^{-1}}^{\bar{i}_{2}}],...],{\mathfrak{L}}^{\bar{i}_{n}}_{{g_{2}}^{-1}}]={\mathfrak{L}}_{g^{\prime}}^{\bar{j}+\bar{1}},$

and so ${\mathfrak{L}}_{g^{\prime}}^{\bar{j}+\bar{1}}\subset I$. Hence
${\mathfrak{L}}_{g^{\prime}}\subset I$ and, taking also into account equation
(21), $I={\mathfrak{L}}$. ∎

######  Theorem 3.1.

Let ${\mathfrak{L}}$ be of maximal length, $\Sigma_{G}$-multiplicative, with
${\mathcal{Z}}({\mathfrak{L}})=0$ and satisfying
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$.
If $|\Sigma_{G}|>2$ and $\Sigma_{G}$ has all of its elements
$\Sigma_{G}$-connected, then the following assertion hold.

  1. 1.

Either ${\mathfrak{L}}$ is gr-simple or

  2. 2.

${\mathfrak{L}}=I\oplus J$ with $I,J$ gr-simple graded ideals of
${\mathfrak{L}}$ satisfying $[I,J]=0$.

###### Proof.

Consider any nonzero graded ideal $I$ of ${\mathfrak{L}}$. Observe that from
equation (22) we can write

$I={\mathfrak{L}}_{1}\oplus(\bigoplus\limits_{g\in\Sigma_{I}}{\mathfrak{L}}_{g})$

where $\Sigma_{I}:=\\{g\in\Sigma_{G}:{\mathfrak{L}}_{g}\subset I\\}$.

Suppose $\Sigma_{I}\neq\Sigma_{G}$. Denote as above

$J:=\bigoplus\limits_{g^{\prime}\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g^{\prime}},$
(32)

being so $J\neq\\{0\\}$. Let us show that the graded subspace $J$ is a graded
ideal of ${\mathfrak{L}}$. We can write

$[J,{\mathfrak{L}}]=[\bigoplus\limits_{g\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g},{\mathfrak{L}}_{1}\oplus(\bigoplus\limits_{g^{\prime}\in\Sigma_{I}}{\mathfrak{L}}_{g^{\prime}})\oplus(\bigoplus\limits_{g^{\prime\prime}\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g^{\prime\prime}})]\subset$
$[\bigoplus\limits_{g\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g},{\mathfrak{L}}_{1}]+[\bigoplus\limits_{g\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g},\bigoplus\limits_{g^{\prime}\in\Sigma_{I}}{\mathfrak{L}}_{g^{\prime}}]+[\bigoplus\limits_{g\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g},\bigoplus\limits_{g^{\prime\prime}\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g^{\prime\prime}}].$
(33)

Taking into account Lemma 3.3-(i), we get that the first summand in equation
(33) vanishes. Let us also observe that Lemma 3.3-(iii) and Lemma 3.4 give us
$[\bigoplus\limits_{g\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g},\bigoplus\limits_{g^{\prime}\in\Sigma_{I}}{\mathfrak{L}}_{g^{\prime}}]=0$.
That is, the second summand in equation (33) is also null and so we have
$[J,{\mathfrak{L}}]\subset[\bigoplus\limits_{g\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g},\bigoplus\limits_{g^{\prime\prime}\in\Sigma_{G}\setminus\Sigma_{I}}{\mathfrak{L}}_{g^{\prime\prime}}].$
Lemma 3.3-(ii) finally shows $[J,{\mathfrak{L}}]\subset J$ and we have that
$J$ is a graded ideal of ${\mathfrak{L}}$.

Let us distinguish two cases. In the first one
${\mathfrak{L}}_{1}\neq\\{0\\}$. Then equation (21) gives us
${\mathfrak{L}}_{1}\subset J$ which contradicts equation (32). Hence
$\Sigma_{I}=\Sigma_{G}$ for any graded ideal $I$ of ${\mathfrak{L}}$ and we
conclude ${\mathfrak{L}}$ is gr-simple in the case
${\mathfrak{L}}_{1}\neq\\{0\\}$.

In the second one, ${\mathfrak{L}}_{1}=\\{0\\}$. We have showed above that we
can write

${\mathfrak{L}}=I\oplus J$ with $J$ a nonzero graded ideal of ${\mathfrak{L}}$
satisfying $[I,J]=0$. (34)

Now, we can prove the gr-simplicity of
$I={\mathfrak{L}}_{1}\oplus(\bigoplus\limits_{g\in\Sigma_{I}}{\mathfrak{L}}_{g})$
by observing that the $\Sigma_{G}$-multiplicativity of ${\mathfrak{L}}$ and
Lemma 3.3-(iii) give us that $\Sigma_{I}$ has all of its elements
$\Sigma_{I}$-connected, that is, $\Sigma_{G}$-connected through
$\Sigma_{G}$-connections contained in $\Sigma_{I}$, and that $I$ is
$\Sigma_{I}$-multiplicative. We also have
${\mathcal{Z}}_{I}(I)=0,({\mathcal{Z}}_{I}(I):=\\{x\in I:[x,I]=0\\}$), as
consequence of equation (34) and ${\mathcal{Z}}({\mathfrak{L}})=0$, and we
clearly have $\dim{\mathfrak{L}}_{g}=1$ for any $g\in\Sigma_{I}$. Since Lemma
3.5 gives us that in case $g\in\Sigma_{I}$ then necessarily
$g^{-1}\notin\Sigma_{I}$, if we consider a nonzero graded ideal
${\widetilde{I}}$ of $I$ we can argue with the fact that $\Sigma_{I}$ has all
of its elements $\Sigma_{I}$-connected, the $\Sigma_{I}$-multiplicativity and
the maximal length of $I$ as usual to get ${\widetilde{I}}=I$. So $I$ is
$gr$-simple. The same argument applies to show $J$ is also a $gr$-simple
graded ideal of ${\mathfrak{L}}$. ∎

It remains to study the case in which $|\Sigma_{G}|\leq 2$. We note that it is
straightforward to describe this class of graded Lie superalgebras in a much
more detailed way than we do in Lemma 3.6. However, it is better for our
purposes the compact description given below.

######  Lemma 3.6.

Let ${\mathfrak{L}}$ be of maximal length, with
${\mathcal{Z}}({\mathfrak{L}})=0$ and satisfying
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$.
If $|\Sigma_{G}|\leq 2$, then one of the following assertions hold.

  1. (1).

${\mathfrak{L}}$ is a gr-simple Lie superalgebra.

  2. (2).

$|\Sigma_{G}|=1$ and ${\mathfrak{L}}$ is a $(2+n)$-dimensional non gr-simple
Lie superalgebra with $n\in\\{1,2,3\\}$, graded as
${\mathfrak{L}}=[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}]\oplus{\mathfrak{L}}_{g}$
with ${\rm dim}({\mathfrak{L}}_{g})=2$ and ${\rm
dim}([{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}])=n$, for some $g\in
G\setminus\\{1\\}$ such that $g=g^{-1}$.

  3. (3).

$|\Sigma_{G}|=2$ and ${\mathfrak{L}}$ is a $3$-dimensional non gr-simple Lie
superalgebra which decomposes as ${\mathfrak{L}}=I\oplus J$, with
$I=[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]\oplus{\mathfrak{L}}_{g}$ a
$2$-dimensional graded ideal of ${\mathfrak{L}}$ and
$J={\mathfrak{L}}_{g^{-1}}$, for some $g\in G\setminus\\{1\\}$ such that
$g\neq g^{-1}$.

  4. (4).

$|\Sigma_{G}|=2$ and ${\mathfrak{L}}$ is a $(4+n)$-dimensional non gr-simple
Lie superalgebra with $n\in\\{0,1,2,3,4\\}$, graded as
${\mathfrak{L}}=[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]\oplus{\mathfrak{L}}_{g}\oplus{\mathfrak{L}}_{g^{-1}}$
with ${\rm dim}({\mathfrak{L}}_{g})={\rm dim}({\mathfrak{L}}_{g^{-1}})=2$ and
${\rm dim}([{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}])=n$, for some $g\in
G\setminus\\{1\\}$ such that $g\neq g^{-1}$.

###### Proof.

Suppose ${\mathfrak{L}}$ is not gr-simple, then there exists a nonzero graded
ideal $I$ of ${\mathfrak{L}}$ such that $I\neq{\mathfrak{L}}$.

In the case $|\Sigma_{G}|=1$, we have that $\Sigma_{G}=\\{g\\}$ for some $g\in
G\setminus\\{1\\}$ such that $g=g^{-1}$. By Lemma 3.1, equation (2) and the
maximal length of ${\mathfrak{L}}$ we have
$0\neq{\mathfrak{L}}_{g}^{\bar{i}}\subset I$ for some
${\bar{i}}\in{\hbox{\hueca Z}}_{2}$. By the maximal length of
${\mathfrak{L}}$, see equation (9), we have two cases to consider. In the
first one ${\rm dim}({\mathfrak{L}}_{g})=1$, being then
${\mathfrak{L}}_{g}={\mathfrak{L}}_{g}^{\bar{i}}$. In this case we can write
${\mathfrak{L}}=[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g}^{\bar{i}}]\oplus{\mathfrak{L}}_{g}^{\bar{i}}$.
Since $0\neq{\mathfrak{L}}_{g}^{\bar{i}}\subset I$, then $I={\mathfrak{L}}$,
(being also $[{\mathfrak{L}}_{g}^{\bar{i}},{\mathfrak{L}}_{g}^{\bar{i}}]\neq
0$ as consequence of ${\mathcal{Z}}({\mathfrak{L}})=0$), a contradiction. From
here, necessarily the second case to consider, in which ${\rm
dim}({\mathfrak{L}}_{g})=2$, happens. This case clearly gives us item (2).

In the case $|\Sigma_{G}|=2$, we have $\Sigma_{G}=\\{g,g^{-1}\\}$ for some
$g\in G$ such that $g\neq g^{-1}$. We also have by Lemma 3.1, equation (2) and
the maximal length of ${\mathfrak{L}}$ that either
$0\neq{\mathfrak{L}}_{g}^{\bar{i}}\subset I$ or
$0\neq{\mathfrak{L}}_{g^{-1}}^{\bar{i}}\subset I$ for some
${\bar{i}}\in{\hbox{\hueca Z}}_{2}$. We can suppose without any loss of
generality that $0\neq{\mathfrak{L}}_{g}^{\bar{i}}\subset I$. By the maximal
length of ${\mathfrak{L}}$, see equation (9), we have again two cases to
consider. In the first one ${\rm dim}({\mathfrak{L}}_{g})={\rm
dim}({\mathfrak{L}}_{g^{-1}})=1$, being then
${\mathfrak{L}}_{g}={\mathfrak{L}}_{g}^{\bar{i}}$ and
${\mathfrak{L}}_{g^{-1}}={\mathfrak{L}}_{g^{-1}}^{\bar{i}}$ for some
$\bar{i}\in{\hbox{\hueca Z}}_{2}$. In this case we can write
${\mathfrak{L}}=[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]\oplus{\mathfrak{L}}_{g}\oplus{\mathfrak{L}}_{g^{-1}}$.
Since $0\neq{\mathfrak{L}}_{g}\subset I$, then
$[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]\subset I$. As we also have
$[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}]\subset{\mathfrak{L}}_{g^{2}}$,
necessarily $[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}]=0$ or
$[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}]\subset{\mathfrak{L}}_{g^{-1}}$. From
here, if $[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}]\neq 0$, then
$0\neq[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}]={\mathfrak{L}}_{g^{-1}}\subset
I$ which would imply $I={\mathfrak{L}}$, a contradiction. Hence
$[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g}]=0.$ Taking into account
${\mathcal{Z}}({\mathfrak{L}})=0$, we get
$[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]\neq 0$. From here ${\rm
dim}({\mathfrak{L}})=3$ and ${\rm dim}({I})=2$ being
$I=[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]\oplus{\mathfrak{L}}_{g}$.
Moreover,
$[[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}],{\mathfrak{L}}_{g^{-1}}]=0$
since in the opposite case ${\mathfrak{L}}_{g^{-1}}\subset I$ and so
$I={\mathfrak{L}}$. This is item (3) in the lemma. Finally, the second case to
consider is the one satisfying ${\rm dim}({\mathfrak{L}}_{g})={\rm
dim}({\mathfrak{L}}_{g^{-1}})=2$, which clearly gives rise to item (4). ∎

######  Theorem 3.2.

Let ${\mathfrak{L}}$ be of maximal length, $\Sigma_{G}$-multiplicative, with
${\mathcal{Z}}({\mathfrak{L}})=0$ and satisfying
${\mathfrak{L}}_{1}=\sum\limits_{g\in\Sigma_{G}}[{\mathfrak{L}}_{g},{\mathfrak{L}}_{g^{-1}}]$.
Then

${\mathfrak{L}}=(\bigoplus\limits_{k\in K}I_{k})\oplus(\bigoplus\limits_{q\in
Q}I_{q}),$

where any $I_{k}$ is a gr-simple graded ideal of ${\mathfrak{L}}$ having its
$G$-support, $\Sigma_{I_{k}}$, with all of the elements
$\Sigma_{I_{k}}$-connected, and any $I_{q}$ is one of the graded ideals
(2)-(4) in Lemma 3.6 satisfying $[I_{q},I_{q^{\prime}}]=0$ for any
$q^{\prime}\in Q$ with $q\neq q^{\prime}$.

###### Proof.

By Corollary 2.1,
${\mathfrak{L}}=\bigoplus\limits_{[g]\in\Sigma_{G}/\sim}I_{[g]}$ is the direct
sum of the graded ideals

$I_{[g]}={\mathfrak{L}}_{{\mathfrak{C}}_{g},1}\oplus{V}_{{\mathfrak{C}}_{g}}=(\sum\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}[{\mathfrak{L}}_{g^{\prime}},{\mathfrak{L}}_{(g^{\prime})^{-1}}])\oplus(\bigoplus\limits_{g^{\prime}\in{\mathfrak{C}}_{g}}{\mathfrak{L}}_{g^{\prime}}),$

having any $I_{[g]}$ as $G$-support, $\Sigma_{I_{[g]}}=[g]$. Observe that we
can write

${\mathfrak{L}}=(\bigoplus\limits_{{\tiny\begin{array}[]{c}[h]\in\Sigma_{G}/\sim;\\\
|[h]|>2\\\
\end{array}}}I_{[h]})\oplus(\bigoplus\limits_{{\tiny\begin{array}[]{c}[h^{\prime}]\in\Sigma_{G}/\sim;\\\
|[h^{\prime}]|\leq 2\\\ \end{array}}}{I}_{[h^{\prime}]}).$ (35)

In order to apply Theorem 3.1 and Lemma 3.6 to each $I_{[h]}$ and
$I_{[h^{\prime}]}$ respectively, observe that any
$I_{[g]},[g]\in\Sigma_{G}/\sim$, is clearly of maximal length and that
${\mathcal{Z}}_{I_{[g]}}(I_{[g]})=0$ as consequence of
$[I_{[g]},I_{[g^{\prime}]}]=0$ if $[g]\neq[g^{\prime}]$, (Corollary 2.1), and
${\mathcal{Z}}({\mathfrak{L}})=0$. Hence, if $|[g]|\leq 2$ then Lemma 3.6
gives us either $I_{[g]}$ is gr-simple or is one of the graded ideals (2)-(4).
In this context, let us denote by

$K_{1}=\\{[g]\in\Sigma_{G}/\sim:|[g]|\leq 2$ and $I_{[g]}$ is gr-simple $\\}$

and by

$\hbox{$Q=\\{[g]\in\Sigma_{G}/\sim:|[g]|\leq 2$ and $I_{[g]}$ is one of the
ideals (2)-(4) in Lemma \ref{cardinal2} $\\}$}.$

Then we can write

$\bigoplus\limits_{{\tiny\begin{array}[]{c}[h^{\prime}]\in\Sigma_{G}/\sim;\\\
|[h^{\prime}]|\leq 2\\\
\end{array}}}{I}_{[h^{\prime}]}=(\bigoplus\limits_{[h^{\prime}]\in
K_{1}}{I}_{[h^{\prime}]})\oplus(\bigoplus\limits_{[h^{\prime}]\in
Q}{I}_{[h^{\prime}]}).$ (36)

If $|\Sigma_{I_{[g]}}|=|[g]|>2$, the $\Sigma_{G}$-multiplicativity of
${\mathfrak{L}}$ and Lemma 3.3-(iii) show that $\Sigma_{I_{[g]}}$ has all of
its elements $\Sigma_{I_{[g]}}$-connected, that is, $\Sigma_{G}$-connected
through $\Sigma_{G}$-connections contained in $\Sigma_{I_{[g]}}$, and that
$I_{[g]}$ is $\Sigma_{I_{[g]}}$-multiplicative. From here, Theorem 3.1 let us
deduce that either $I_{[g]}$ is gr-simple or $I_{[g]}=I_{I_{[g]}}\oplus
J_{I_{[g]}}$ with $I_{I_{[g]}},J_{I_{[g]}}$ gr-simple graded ideals of
$I_{[g]}$ satisfying $[I_{I_{[g]}},J_{I_{[g]}}]=0$. Now let us denote by

$K_{2}=\\{[g]\in\Sigma_{G}/\sim:|[g]|>2$ and $I_{[g]}$ is gr-simple $\\}$

and by

$\hbox{$K_{3}=\\{[g]\in\Sigma_{G}/\sim:|[g]|>2$ and $I_{[g]}=I_{I_{[g]}}\oplus
J_{I_{[g]}}$ with $I_{I_{[g]}},J_{I_{[g]}}$ gr-simple $\\}$}.$

We can assert

$\bigoplus\limits_{{\tiny\begin{array}[]{c}[h]\in\Sigma_{G}/\sim;\\\
|[h]|>2\\\ \end{array}}}{I}_{[h]}=(\bigoplus\limits_{[h]\in
K_{2}}{I}_{[h]})\oplus(\bigoplus\limits_{[h]\in
K_{3}}I_{I_{[h]}})\oplus(\bigoplus\limits_{[h]\in K_{3}}J_{I_{[h]}}).$ (37)

Taking now into account equations (35), (36) and (37), we get the
decomposition

${\mathfrak{L}}=(\bigoplus\limits_{k\in K}I_{k})\oplus(\bigoplus\limits_{q\in
Q}I_{q}),$

where any $I_{k}$ is a gr-simple graded ideal of ${\mathfrak{L}}$ and any
$I_{q}$ is one of the graded ideals (2)-(4) in Lemma 3.6, satisfying
$[I_{q},I_{q^{\prime}}]=0$ for any $q^{\prime}\in Q$ with $q\neq q^{\prime}$.
∎

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