  The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite
Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$
consisting of certain minimal length coset representatives of $W_J$ in $W$. In
this note we show that (with the exception of a small number of cases arising
from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of
an element $w \in W$ is in a similar way a semidirect product of the
centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement
isomorphic to the normalizer complement $N_J$.
