Weyl group of a restricted root system

Question

What is the order of the Weyl group of the restricted root system of the real Lie algebra $\mathfrak g= \mathfrak{so}(p,q)$?

More precisely, $\mathfrak g= \mathfrak k \oplus \mathfrak p$ and let $\mathfrak a$ be the maximal abelian subalgebra in $\mathfrak p$. We have the restricted root system $\Phi(\mathfrak g,\mathfrak a)$, where the roots are elements $\lambda\in\mathfrak a^*$ such that the following space is nonzero: $$ \{X\in\mathfrak g : (\mathrm{ad} H)X=\lambda(H) X \text{ for all }X\in\mathfrak a\}. $$

What is the order of the Weyl group of this root system associated to $\mathfrak g=\mathfrak{so}(p,q)$?

Answer 1

If $p = q$ you have the split form of even degree, so the restricted root system is the whole root system, of type $D_p$, and its Weyl group has order $2^{p-1} \cdot p!$.

In all other cases, if I read the tables correctly, the restricted root system is of type $B_r$ for $r = \min(p,q)$ (the Witt index of the form) and therefore its Weyl group has order $2^r \cdot r!$.