Real and complex Clifford algebra classification with explicit signature

By [F1], a nondegenerate real form of signature $(p,q)$ has an orthogonal basis with $p$ squares $+1$ and $q$ squares $-1$. In the quotient convention [F7], the linear isometry from this standard form to the given form induces a Clifford homomorphism by [F5]; applying [F5] to the inverse isometry gives its inverse, since both composites fix every canonical generator. Thus it suffices to classify the standard ${\mathrm{C}\mathrm{l}}_{p,q}$.

Let $m=\min (p,q)$. Repeatedly applying the first recurrence in [F2] removes one positive and one negative generator at a time, giving ${\mathrm{C}\mathrm{l}}_{p,q}\cong {\mathrm{C}\mathrm{l}}_{p-q,0}\otimes M_{2^q}(\mathbb{R})$ when $p\ge q$, and ${\mathrm{C}\mathrm{l}}_{p,q}\cong {\mathrm{C}\mathrm{l}}_{0,q-p}\otimes M_{2^p}(\mathbb{R})$ when $q>p$. If the remaining pure dimension is $8t+s$ with $0\le s<8$, [F2] replaces its $8t$-part by a factor $M_{{16}^t}(\mathbb{R})$.

Let $B$ be a nondegenerate symmetric complex bilinear form on a positive-dimensional complex vector space. Some vector $v$ has $B(v,v)\mathrm{\ne }0$: otherwise polarization, valid because $2$ is invertible by [F8], would give $B=0$, contradicting nondegeneracy. By [F4], rescale $v$ so that $B(v,v)=1$. Then every $x$ decomposes as $x=B(x,v)v+(x-B(x,v)v)$ with the second term in $v^{\perp }$, so $V=\mathbb{C}v\oplus v^{\perp }$ and $\dim v^{\perp }=n-1$. The restriction to $v^{\perp }$ is nondegenerate, since a vector in its radical is orthogonal to both $v^{\perp }$ and $v$, hence to all of $V$. Induction therefore gives an orthogonal basis with every square $+1$, so all nondegenerate complex forms of dimension $n$ are isometric to the standard one.

If $n=0$, the PBW basis from [F6] consists only of the unit, so the scalar map $\mathbb{C}\to {\mathrm{C}\mathrm{l}}_0(\mathbb{C})$ is an isomorphism: it is nonzero and hence injective, and both sides have complex dimension one.

For $p\ge q$, the pure positive column in [F2] supplies, for residues $s=0,\dots ,7$, coefficient types $\mathbb{R}$, $\mathbb{R}\oplus \mathbb{R}$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{H}\oplus \mathbb{H}$, $\mathbb{H}$, $\mathbb{C}$, with matrix-size exponents respectively $m+4t$, $m+4t$, $m+4t+1$, $m+4t+1$, $m+4t+1$, $m+4t+1$, $m+4t+2$, $m+4t+3$. Since $n=8t+s+2m$, these are exactly the exponents in the table of the Statement. For $q>p$, apply the negative pure column of [F2] with $q-p=8t+s$ and $0\le s<8$; its eight residue rows give respectively residues $r=0,7,6,5,4,3,2,1$ and matrix-size exponents $m+4t$, $m+4t$, $m+4t$, $m+4t$, $m+4t+1$, $m+4t+2$, $m+4t+3$, $m+4t+3$. Substituting $n=8t+s+2m$ again gives precisely the displayed real table, including the split rows.

Suppose $n>0$. The real generator map into the complex Clifford algebra satisfies the same squares, so [F5] extends it to a real-algebra homomorphism $f:{\mathrm{C}\mathrm{l}}_{n,0}(\mathbb{R})\to {\mathrm{C}\mathrm{l}}_n(\mathbb{C})$. The real-bilinear map $(a,z)\mapsto f(a)z$ induces by [F9] a linear map $\mathrm{\Psi }:{\mathrm{C}\mathrm{l}}_{n,0}(\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}\to {\mathrm{C}\mathrm{l}}_n(\mathbb{C})$, $a\otimes z\mapsto f(a)z$. It is multiplicative because complex scalars are central in the target: $\mathrm{\Psi }((a\otimes z)(b\otimes w))=f(ab)zw=f(a)zf(b)w$. Its image contains the complex scalars and all complex Clifford generators, so it is surjective; both sides have complex dimension $2^n$ by [F6], making it an isomorphism. For a real matrix factor, [F3] gives $M_s(D){\otimes }_{\mathbb{R}}\mathbb{C}\cong M_s(D{\otimes }_{\mathbb{R}}\mathbb{C})$. The coefficient algebra changes as follows: $\mathbb{R}\otimes \mathbb{C}\cong \mathbb{C}$ by the tensor-unit map [F12], which preserves products; $\mathbb{C}\otimes \mathbb{C}\cong \mathbb{C}\oplus \mathbb{C}$ and $\mathbb{C}\otimes \mathbb{H}\cong M_2(\mathbb{C})$ by [F3]; and $\mathbb{H}\otimes \mathbb{C}\cong \mathbb{C}\otimes \mathbb{H}$ by the flip [F10], which is multiplicative since it sends $(h\otimes z)(h^{\mathrm{\prime }}\otimes z^{\mathrm{\prime }})$ to $z{z}^{\mathrm{\prime }}\otimes h{h}^{\mathrm{\prime }}=(z\otimes h)(z^{\mathrm{\prime }}\otimes h^{\mathrm{\prime }})$. The direct-sum isomorphism [F11] preserves componentwise products, so split rows remain two factors. Thus real residue rows $r=0,2$ remain single $\mathbb{C}$-matrix factors of the same size; rows $r=4,6$ replace each $\mathbb{H}$ by $M_2(\mathbb{C})$, doubling the matrix size; rows $r=1,5$ remain split into two factors, with the $r=5$ matrix size doubled; and rows $r=3,7$ split because $\mathbb{C}\otimes \mathbb{C}\cong \mathbb{C}\oplus \mathbb{C}$. Substituting the real table sizes in [F2] gives one factor $M_{2^{n\mathrm{/}2}}(\mathbb{C})$ for even $n$ and two factors $M_{2^{(n-1)\mathrm{/}2}}(\mathbb{C})$ for odd $n$. Together with step 1.3, this classifies every nondegenerate complex form.