Solution
= Solution
Write the basic construction as
$$
U(W,K)=(W\times K)/\sim,
$$
where $(w,x)\sim(w',x)$ exactly when $w^{-1}w'$ belongs to the subgroup generated by the mirrors containing $x$. Let $v_T$ be the vertex of $K$ corresponding to a <spherical subset of a Coxeter system> $T$. Its stabilizer is $W_T$. Therefore
$$
\operatorname{Stab}_W([w,v_T])
=wW_Tw^{-1}.
$$
Every conjugate of every spherical standard parabolic subgroup consequently occurs as a vertex stabilizer.