Low-dimensional real Clifford signatures

For ${\mathrm{C}\mathrm{l}}_{1,0}$, send its positive generator to $(1,-1)\in \mathbb{R}\oplus \mathbb{R}$; its square is $(1,1)$. The image contains the linearly independent vectors $(1,1)$ and $(1,-1)$, so the homomorphism is onto. Every word in one generator reduces using $e^2=1$ to a linear combination of $1,e$, so the source has dimension at most two; the target has dimension two, hence the map is an isomorphism. For ${\mathrm{C}\mathrm{l}}_{0,1}$, send its negative generator to $i\in \mathbb{C}$; $i^2=-1$, and $1,i$ span the target. Again every word reduces to $1,e$, so the surjection between spaces of dimension at most two and two is an isomorphism.

Put $A=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, $B_{+}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, and $B_{-}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Direct multiplication gives $A^2=B_{+}^2=I$, $B_{-}^2=-I$, and $A{B}_{+}=-B_{+}A$, $A{B}_{-}=-B_{-}A$. Thus $A,B_{+}$ define a homomorphism from ${\mathrm{C}\mathrm{l}}_{2,0}$ and $A,B_{-}$ one from ${\mathrm{C}\mathrm{l}}_{1,1}$. In the positive case the four matrices are $I,A,B_{+},B_{-}$; in the mixed case they are $I,A,B_{-},B_{+}$. For any real matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, its unique expansion in $I,A,B_{+},B_{-}$ has coefficients $(a+d)\mathrm{/}2,(a-d)\mathrm{/}2,(b+c)\mathrm{/}2,(b-c)\mathrm{/}2$, so these four matrices are a basis and each displayed list spans. The defining relations reduce every word to a linear combination of $1,e_1,e_2,e_1{e}_2$, so each source has dimension at most four; surjectivity onto the four-dimensional matrix algebra makes both maps isomorphisms.

In $\mathbb{H}$, the elements $i,j$ square to $-1$ and anticommute. They therefore define a homomorphism from ${\mathrm{C}\mathrm{l}}_{0,2}$; its image contains $1,i,j,ij=k$, which is a basis of $\mathbb{H}$. Every word reduces to a linear combination of $1,e_1,e_2,e_1{e}_2$, so the source has dimension at most four. The surjection onto the four-dimensional $\mathbb{H}$ is an isomorphism.