A subgroup of the Weyl group 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.
 A: As indicated in my comments, the 1968 book Simple Groups of Lie Type by R.W. Carter has a good elementary treatment of your question (to which the answer is yes) in Chapter 13.   All of this goes back pretty far in the history of Lie theory, with the Weyl group and root system arising from a simple Lie algebra (or Lie group) or from a simple algebraic group.   Typically the discussion of symmetries of the Dynkin diagram occurs along with specific constructions in the Lie algebra or associated group.   But Carter's exposition is clear and detailed, showing how to pass from the original Weyl group to its subgroup commuting with the given diagram symmetry.   In particular, this subgroup has a natural set of generators as in your question.
While there are many treatments in textbooks or lecture notes, one online source may be useful: the 1967-68 Yale lecture notes on Chevalley groups by Steinberg 
here.  See his section 11, where he starts with standard examples and then treats the general theory.   In all such sources, notation varies quite a bit but the ideas are pretty much the same.
