  Let $(W,S)$ be a Coxeter system, let $S=I \dot{\cup} J$ be a partition of $S$
such that no element of $I$ is conjugate to an element of $J$, let
$\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let
$\widetilde{W}$ be the subgroup of $W$ generated by $\widetilde{J}$. We show
that $W=\widetilde{W} \rtimes W_I$ and that $(\widetilde{W},\widetilde{J})$ is
a Coxeter system.
