Real Pin and full-even Spin images, kernels, and definite compactness

Let $A\in O(V,B)$. By [F1], write $A=r_{u_1}\cdots r_{u_k}$. Replacing each anisotropic $u_j$ by $u_j\mathrm{/}\sqrt{\mathrm{\mid }Q(u_j)\mathrm{\mid }}$ preserves its reflection and gives a Pin generator; [F2] shows their product maps to $A$. Thus $\rho$ maps Pin onto $O(V,B)$.

Each normalized vector has determinant $-1$ under $\rho$. Therefore an even product maps into $SO(V,B)$, and every $A\in SO(V,B)$ has an even reflection factorization by [F1]; its normalized lift lies in full Spin. Hence $\rho$ maps full Spin onto $SO(V,B)$.

The kernel of either restriction is contained in the scalar kernel ${\mathbb{R}}^{\times }$ by [F2]. If a scalar $\lambda$ lies in Pin, [F3] gives $N(\lambda )={\lambda }^2\in \{1,-1\}$. Since $\lambda$ is real, ${\lambda }^2>0$, so ${\lambda }^2=1$ and $\lambda =\pm 1$. Conversely both scalars belong to Pin and full Spin and act trivially. This proves exactness with kernel $\{1,-1\}$.

Scalars are central in the Clifford algebra, so the two kernels are central. For a definite form with $n>0$, the normalized-vector set $U=\{u:\mathrm{\mid }Q(u)\mathrm{\mid }=1\}$ is a compact sphere. Given $x\in \mathrm{Pin}$, factor ${\rho }_x$ by [F1] into at most $n$ reflections and lift them to a normalized-vector product $y$. Then $x{y}^{-1}$ is scalar by [F2], and step 1.3 makes it $\pm 1$; hence every Pin element is $\pm$ a product of at most $n$ vectors. If $x$ is full-even Spin, its image has determinant $1$, so [F1] gives an even reflection factorization and the same argument gives an even product of at most $n$ vectors. Each fixed-length product is the continuous image of a finite product of $U$, hence compact; the groups are finite unions of these images. In dimension zero both are the finite group $\{1,-1\}$.