Even subalgebras of real and complex Clifford algebras

Suppose first that $p>0$. Let $e_1$ be a positive generator, let $e_2,\dots ,e_p$ be the other positive generators, and let $f_1,\dots ,f_q$ be the negative generators. Set $u_i=e_1{e}_{i+1}$ for $1\le i<p$ and $v_j=e_1{f}_j$ for $1\le j\le q$. These elements are even by [F1]. Using $e_1^2=1$ and the orthogonal anticommutation in [F3], each $u_i^2=-1$, each $v_j^2=+1$, and every two distinct elements in the combined list anticommute. Thus the $v_j$ are the $q$ positive generators and the $u_i$ the $p-1$ negative generators for the defining relations of ${\mathrm{C}\mathrm{l}}_{q,p-1}$; [F4] gives an algebra homomorphism ${\mathrm{C}\mathrm{l}}_{q,p-1}\to {\mathrm{C}\mathrm{l}}_{p,q}^0$.

If $p=0$, choose a negative generator $f_1$ and set $u_j=f_1{f}_{j+1}$ for $1\le j<q$. Each $u_j$ is even by [F1], squares to $-1$, and anticommutes with the other $u_l$ by [F2]–[F3]. For distinct remaining generators $f_i,f_j$, the identity $(f_1{f}_i)(f_1{f}_j)=f_i{f}_j$ shows that these pair products generate every even ordered monomial; when $q=1$ there are no $u_j$ and the only even monomial is the unit. Thus [F4] gives a surjection ${\mathrm{C}\mathrm{l}}_{0,q-1}\to {\mathrm{C}\mathrm{l}}_{0,q}^0$.

The image contains every product of two distinct standard generators: $e_1{e}_i$ and $e_1{f}_j$ are generators of the image, while for $i,j>1$, $(e_1{e}_i)(e_1{e}_j)=-e_i{e}_j$, and the same identity holds for a pair of negative generators or one positive and one negative generator. Every even ordered monomial is a product of such pairs, so the homomorphism is surjective; if $p=1,q=0$, the only even monomial is the unit and the scalar map is already surjective.

Both algebras in step 1.2 have dimension $2^{q-1}$ by [F5], so the surjection is an isomorphism. This includes $q=1$, when [F7] identifies the source with $\mathbb{R}$ and the even part of ${\mathrm{C}\mathrm{l}}_{0,1}$ is its scalar subalgebra.

By [F5], ${\mathrm{C}\mathrm{l}}_{p,q}^0$ has dimension $2^{p+q-1}$, equal to the dimension of ${\mathrm{C}\mathrm{l}}_{q,p-1}$. The surjective map of finite-dimensional vector spaces is therefore injective, proving the first isomorphism.

For the complex standard form, choose the first generator $e_1$ and set $u_j=e_1{e}_{j+1}$. The construction and checks in steps 1.1, 2.1, and 3.1 show that the even subalgebra is generated by $n-1$ pairwise anticommuting elements of square $-1$ and has the PBW dimension $2^{n-1}$; hence it is ${\mathrm{C}\mathrm{l}}_{0,n-1}(\mathbb{C})$. Multiplying each of these generators by $i\in \mathbb{C}$ changes its square to $+1$ and gives a surjective homomorphism from the standard complex Clifford algebra ${\mathrm{C}\mathrm{l}}_{n-1}(\mathbb{C})$; the equal PBW dimensions make it an isomorphism. The argument includes $n=1$, where the even subalgebra is $\mathbb{C}$.