Real Clifford group, Pin, and the two Spin subgroups

Let $(V,B)$ be a finite-dimensional nondegenerate real quadratic space with $Q(v)=B(v,v)$, and let $\mathrm{C}\mathrm{l}(V,Q)$ use $v^2=Q(v)1$. Use the grading involution $\alpha$ and Clifford conjugation $\bar{x}$ from Clifford involutions and the norm of a vector product. Let ${\mathrm{C}\mathrm{l}}^0$ and ${\mathrm{C}\mathrm{l}}^1$ be the even and odd parts for word-length parity. Define the unrestricted twisted Clifford normalizer as the set

$\mathrm{\Gamma }(V,B)=\{x\in \mathrm{C}\mathrm{l}(V,Q{)}^{\times }:\alpha (x)v{x}^{-1}\in V\text{ for every }v\in V\},{\rho }_x(v)=\alpha (x)v{x}^{-1}\mathrm{.}$

Define the normalized anisotropic vectors $U(V,Q)=\{u\in V:\mathrm{\mid }Q(u)\mathrm{\mid }=1\}$. For $\dim V>0$, Pin$(V,Q)$ is the subgroup of $\mathrm{C}\mathrm{l}(V,Q{)}^{\times }$ generated by $U(V,Q)$; for $V=0$, set Pin$(0)=\{1,-1\}$. Define

${\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(V,Q)=\mathrm{Pin}(V,Q)\cap {\mathrm{C}\mathrm{l}}^0,{\mathrm{Spin}}_{N=1}(V,Q)=\{x\in {\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(V,Q):N(x)=1\}\mathrm{.}$

For definite signature, the conventional names Pin${}^{+}(n)$ and Pin${}^{-}(n)$ mean the groups in ${\mathrm{C}\mathrm{l}}_{n,0}$ and ${\mathrm{C}\mathrm{l}}_{0,n}$, respectively. In mixed signature the subscript “full” is retained because the full even subgroup and the norm-one subgroup can have different images.

The unrestricted normalizer's closure and homogeneity are local proof obligations in this pair. The scalar $-1$ is included explicitly in dimension zero.