Low-dimensional real Clifford algebra models

For ${\mathrm{C}\mathrm{l}}_{1,0}$, send its positive generator to $(1,-1)\in \mathbb{R}\oplus \mathbb{R}$; every multiple $xe$ then squares to $x^2(1,1)$, so [F2] gives a homomorphism. Its image contains $(1,1)$ and $(1,-1)$, which span the target over $\mathbb{R}$, and [F3] gives dimension $2$ on the source, so the map is an isomorphism. For ${\mathrm{C}\mathrm{l}}_{0,1}$, send its negative generator to $i\in \mathbb{C}$; every multiple $xe$ squares to $-x^2$, and the images $1,i$ are a basis, so [F2]–[F3] give an isomorphism.

Let $A=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, $B_{+}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, 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 $A{B}_{+}=-B_{+}A$, $A{B}_{-}=-B_{-}A$. Hence $(xA+y{B}_{+}{)}^2=(x^2+y^2)I$ and $(xA+y{B}_{-}{)}^2=(x^2-y^2)I$ for all real $x,y$, so [F2] gives the required homomorphisms from ${\mathrm{C}\mathrm{l}}_{2,0}$ and ${\mathrm{C}\mathrm{l}}_{1,1}$. In each case the images of $1$, the two generators, and their product are the four linearly independent matrices $I,A,B_{+},B_{-}$ in some order, so the maps are surjective; [F3] gives dimension $4$ on each source, making both maps isomorphisms.

Send the two negative generators of ${\mathrm{C}\mathrm{l}}_{0,2}$ to $i,j\in \mathbb{H}$. They square to $-1$ and anticommute, so every linear combination $xi+yj$ squares to $-(x^2+y^2)$ and [F2] gives a homomorphism. Its image contains $1,i,j,ij=k$, the basis in [F4]; [F3] gives dimension $4$ on the source, so this homomorphism is an isomorphism.