Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2. Let $Q=Q(D)$ denote the root lattice of $D$. Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is generated by reflections $r_\alpha$ for $\alpha\in D$. The automorphism $\nu$ acts on $Q$. Let $W_0$ denote the centralizer of $\nu$ in $W\subset {\rm Aut}\, Q$.

I want to understand this group $W_0$. Let $D^\nu$ denote the subset of $\nu$-fixed vertices in $D$.
For $\beta\in D^\nu$ we have $r_\beta\in W_0$.
I assume that for all $\gamma\in D\smallsetminus D^\nu$, the vertices $\gamma$ and $\nu(\gamma)$ are *not* connected by an edge
(thus I exclude the case $D={\bf A}_{2n}$).
Then $r_\gamma$ and $r_{\nu(\gamma)}$ commute, and we have $r_\gamma r_{\nu(\gamma)}\in W_0$.

Question.Is it true that $W_0$ is generated by $r_\beta$ for $\beta\in D^\nu$ and by $r_\gamma r_{\nu(\gamma)}\in W_0$ for $\gamma\in D\smallsetminus D^\nu$?

I am interested in the case $D={\bf D}_n$, but I would prefer to get a classification-free answer.

Simple Groups of Lie Typemay be a good source for an affirmative answer to your question, but I'd need to check more carefully. There are also accounts by Tits, Satake, etc. $\endgroup$ – Jim Humphreys Aug 23 '13 at 13:20