spatial dimensions) may be naturally incorporated into the algebra of symmetries, $B^0$. When studying either insulators or superconductors which correspond to gapped Hamiltonians, $\mathcal{H}(\mathbf{k})$, the mass matrix, $M$, which gaps the spectrum, may be spectrally flattened (i.e. ${M^2=-1}$), and, together with the algebra of symmetries, $B^0$, form an extended algebra, $B$. Classifying all gapped Dirac Hamiltonians is thus equivalent to classifying all Hilbert space actions [@abramovici2012clifford] of the extended algebra $B$ which are compatible with the action of $B^0$; see Figs. \[fig:diag1\] and \[fig:diag2\]. One finds a Morita equivalent Clifford algebra structure [@kitaev2009periodic; @atiyah1964clifford; @Vela2002Central; @cliff] for both algebras, ${B^0={{\mathrm{Cl}}}_{p,q}}$ and ${B={{\mathrm{Cl}}}_{p+1,q}}$. Moreover, one may also identify the extended algebra as a ${\mathbb{Z}}_2$-graded algebra (superalgebra) $B=B^0\oplus B^1$ whose even part is the algebra of symmetries $B^0$; see Sec. \[sec:graded\]. The space of Dirac Hamiltonians corresponding to the extension ${{{\mathrm{Cl}}}_{q+6,d}\hookrightarrow{{\mathrm{Cl}}}_{q+7,d}}$ is stably homotopic to a Cartan symmetric space ${\mathcal{R}}_{q-d}$ which is the classifying space [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @Hasan2010Colloquium; @moore2010birth; @franz2013topological; @witten2015three] of the AZ symmetry class $q$ in $d$ dimensions. Every stable topological phase is therefore characterized by an invariant in the group $\pi_0({\mathcal{R}}_{q-d})$; see Sec. \[sec:ext\]. This structure is summarized in the periodic table [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @Hasan2010Colloquium; @moore2010birth; @franz2013topological; 