On a parabolic symmetry of finite Coxeter groups

Let (W, S) be a finite Coxeter system, and let J subset of or equal to S. Any w is an element of W has a unique factorization w=w(J)w(J), where w(J) belongs to the parabolic subgroup W-J generated by J, and w(J) is of minimal length in the coset wW(J). It is shown here that w(I) = w(J) if and only if w(I) = w(J), for all I, J subset of or equal to S. Furthermore, a similar symmetry property in arbitrary (W-I, W-J)-double cosets is conjectured, which links this result to the Solomon descent algebra of W.