While reading about the structure of Clifford algebra, there were two facts listed as bullet points about the center of Clifford algebra based on the parity of the dimension of the underlying vector space that I'm now curious about.
Say you have a finite dimensional vector space $V$ over a field $K$, such that $\operatorname{char} K\neq 2$, and $G$ is a symmetric bilinear form. Let $\mathrm{Cl}_G(V)$ denote the corresponding Clifford algebra.
Apparently, if $\dim V$ is even, then the center of $\mathrm{Cl}_G(V)$ coincides with $K$. However, if $\dim V$ is odd, then the center is actually a $2$-dimensional vector space over $K$.
These seem like good interesting facts to know, but I couldn't find an authoritative reference containing a proof of either of them. Does anybody here have a nice proof of either one/both I could read over? I'd appreciate it very much, thanks.