algebra of $2\times 2$ real matrices, $M_2(\mathbb{R})$, is generated over $\mathbb{R}$ by $\gamma_1,\gamma_2$ subject to the relations ${\gamma_1^2=\gamma_2^2=1}$ and ${\gamma_1 \gamma_2=-\gamma_2 \gamma_1}$. We can define the $\mathbb{Z}_2$-grading by making both $\gamma_1$ and $\gamma_2$ odd, in which case $\gamma_1 \gamma_2$ will be even, and the even part would be ${\mathbb{R}\langle \gamma_1 \gamma_2\rangle \cong \mathbb{C}}$; this corresponds to the identification ${M_2({\mathbb{R}})\cong{{\mathrm{Cl}}}_{0,2}}$. However, we could also define the grading by making $\gamma_1$ odd and $\gamma_2$ even, in which case the even part would be ${\mathbb{R}\langle \gamma_2 \rangle\cong \mathbb{R}\oplus \mathbb{R}}$; this corresponds to the identification ${M_2({\mathbb{R}})\cong{{\mathrm{Cl}}}_{1,1}}$.

Given two $\mathbb{Z}_2$-graded algebras $A$ and $A'$ generated by homogeneous elements, the graded tensor product $A \hat{\otimes} A'$ is defined to be the $\mathbb{R}$-algebra generated by the generators of $A$ and $A'$ put together such that they satisfy their former relations, plus the following: every even generator of $A$ commutes with all the generators of $A'$, and vice versa, and for odd generators ${a\in A^1}$ and ${a'\in A'^1}$ we have ${aa'=-a'a}$.

Specifically, the ${\mathbb{Z}}_2$-graded tensor product [@Vela2002Central] of two Clifford algebras is simply given by $$\label{cliffprod}
{{\mathrm{Cl}}}_{p_1,q_1}\hat{\otimes}{{\mathrm{Cl}}}_{p_2,q_2}={{\mathrm{Cl}}}_{p_1+p_2,q_1+q_2}.$$

Given an algebra $A$, the algebra of $n\times n$ matrices with entries in $A$ is denoted by $M_n(A)$. Recall that by 