Need explanation of the spinor norm
Wikipedia and Groupprops gave a definition, but they didn't elaborate, so I don't understand, and there aren't cited sources on them.
Is there any online sources that have proof on their basic property, such that the spinor norm is well-defined, or that its kernel is the derived subgroup of SO and is normal?
Why is it called "spinor" norm, does it have any relationship to spinor?
What's its geometrical meaning? I understand that this map is trivial for R and C so "geometrical" meaning my be hard, but hopefully there is something like that for say Q.
I'm not sure about an online source, but you can check section 9.3 of Quadratic and Hermitian Forms by Scharlau, there are a reasonable amount of details.
It is called the spinor norm because it is actually naturally defined on the spinor group. Indeed, you have a natural involution $x\mapsto \sigma(x)$ on the Clifford algebra $C(V,q)$ of a quadratic space $(V,q)$ (which is characterized by the fact that it is the identity on $V$), and thus you have a "norm" $N: C(V,q)\to C(V,q)$ given by $x\mapsto x\sigma(x)$. (This may be reminiscent of the quaternion norm.) Then if you restrict $N$ to the spinor group $\Gamma(V,q)\subset C(V,q)$, you actually get a group morphism $\Gamma(V,q)\to K^*$, which induces the spinor norm $O(V,q)\to K^*/K^{*2}$.
Note that it is not true that the spinor norm is trivial over $\mathbb{R}$; it is trivial for the usual scalar product, but there are other quadratic forms over $\mathbb{R}$, for which the spinor norm need not be trivial.