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.


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*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.
 A: 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. 
A: There is an easier characterization of the Spinor norm, but it only works for Isogenic quadratic forms in dimension $2$ or $3$. Suppose $Q$ is a quadratic form on $V=K^3$, given by the matrix:
$$\begin{pmatrix}0 & 1 & 0 \\1&0&0\\0&0&1\end{pmatrix}$$
Consider the zero locus: $Z(Q)^*=\{v\in V\setminus\{0\}: Q(v)=0\}$. This set has an equivalence relation given by $z_1\sim z_2$ if one of the following conditions is satisfied:
a) $z_1=\lambda z_2$ for a certain $\lambda\in K^{*2}$
b) $-2\langle z_1,z_2\rangle\in K^{*2}$. 

The quotient space $Z(Q)^*/\sim$ is given by:
$$\{[\lambda e_1]: \lambda\in K^*/K^{*2}\}$$
The Spinor norm reflects how $O(V,Q)$ acts on $Z(Q)^*/\sim$.
For the general case, Captain Lama covers it pretty well.
EDIT: This actually has a nice geometric interpretation on $K=\mathbb{R}$. On that field, we can partition $Z(Q)$ into: $$S_1=\{(x,y,z)\in Z(Q): x\leqslant 0, y\geqslant 0\}\\ S_2=\{(x,y,z)\in Z(Q): x\geqslant 0, y\leqslant 0\}$$
Transformations with Spinor norm $-1$ permute between those sets, whereas transformations with Spinor norm $1$ keep those sets intact.
