the renowned Artin-Wedderburn theorem, every semi-simple algebra decomposes uniquely as a direct sum of matrix algebras over division algebras. Two semi-simple algebras are said to be “Morita equivalent" if they decompose as direct sums of matrix algebras (of possibly different dimensions) over the same division algebras. For example, $\mathbb{R}\oplus \mathbb{C}$ is Morita equivalent to $M_2(\mathbb{R}) \oplus M_3({\mathbb{C}})$. When the algebras are also $\mathbb{Z}_2$-graded, we say the algebras are equivalent if they are equivalent and also their even parts are Morita equivalent.

### (${\mathbb{Z}}_2$-graded) Algebra Extensions {#sec:ext}

By the Bott periodicity [@atiyah1964clifford], all real Clifford algebras [@cliff] are Morita equivalent to eight prescribed algebras; see Table \[tab:per\]. Their parametrization is a matter of choice, and we follow Kennedy and Zirnbauer \[\] to consider the algebra ${B=\langle{{\mathrm{Cl}}}_{q+6+1,d},\Pi(G)\rangle}$ for a given $q \in \{0,1,\dots,7\}$. The presentation of ${{\mathrm{Cl}}}_{q+6+1,d}$ we consider is the following: write ${\gamma_1,\gamma_2,\dots,\gamma_d}$ for the Dirac gamma matrices, ${x_1,\dots,x_{q+6}}$ for the non-spatial symmetries, and $x_0$ for the spectrally flattened mass matrix. These generators satisfy ${\gamma_i^2=1,~x_i^2=-1}$. We write $y_1,\dots,y_{q+7+d}$ for the generators $x_1,\dots,x_{q+6};x_0;\gamma_1,\dots,\gamma_d$, and we refer to the generators this way in statements which hold for all the generators.

Let us look at a Hilbert space, $E$, which is a $B$-module and 