Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ relative to the given generating set $S$, which can be expressed as the $h/2$-power of a well-chosen Coxeter element when the Coxeter number $h$ is even (or a slightly modified expression if $h$ is odd).

Assume now that $W$ is a Weyl group (the others being a little more complicated to study). A standard fact is that $w_o = -1$ just when the type of $W$ in the classification is not ADE; more precisely, is different from $A_\ell (\ell \geq 2), D_\ell (\ell \geq 4), E_6$. When $w_o =-1$, its centralizer is obviously $W$, but otherwise is a proper subgroup.

A recent question here suggests to me a possible uniform treatment of such centralizers in terms of foldings of Coxeter graphs. But I'm not sure how far this is supported by the literature, or exactly how it might work for type $A_\ell$ with $\ell$ even --- then $W$ is the symmetric group $S_{\ell+1}$ and $h=\ell+1$ is odd. (I'm also unsure about the non-crystallographic types).

Note that folding the graph of (say) type $E_6$ yields the graph $F_4$. The Coxeter numbers are always the same when such foldings of ADE graphs occur, e.g., $h=12$ for both $E_6$ and $F_4$. Moreover, $w_o$ is folded into the corresponding longest element $-1$ in accordance with the above expressions as powers of Coxeter elements if $h$ is even. The question cited suggests that the "folded" Weyl group might always embed in $W$ as the precise centralizer of the original $w_o$.

Is this true, and is there a reference? (Further, what can be said in the non-crystallographic case?)