I have been reading Lawson & Michelsohn's Spin Geometry, and there is a remark about the decomposition of $\text{Cl}_{r,s}$ in terms of $\mathbb Z_2$-graded tensor product below the propostion $3.2$
There is an isomorphism $$\text{Cl}_{r,s} \cong \text{Cl}_1 \hat \otimes \cdots \hat \otimes \text{Cl}_1 \hat \otimes \text{Cl}_1^\ast \hat \otimes \cdots \hat \otimes \text{Cl}_1^\ast$$ where $\text{Cl}_1$ appears $r$ times and $\text{Cl}_1^\ast$ appears s times on the right.
($\hat\otimes$ denotes the multiplication in $\mathbb Z_2$-graded algebra .)
The remark is that
Proposition $3.2$ is, however, not so useful if we wish to represent $\text{Cl}_{r,s}$ as a matrix algebra.
But I learned from another source that one can actually somehow use this result in the representation. My question is that, suppose one can find a representation in terms of graded tensor products instead of ungraded ones with more efforts, would it be any better?