2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164408Let $(W,S)$ 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.28 pages, one table. We have added some comments on parabolic subgroups, double cosets representatives, finite and affine Weyl groups, invariant theory, Solomon descent algebraGroup Theory20F55text