  In this paper, we study the decomposition of Bruhat intervals in a Coxeter
group with respect to cosets of a parabolic subgroup. Our main result is that
the intersection of a lower Bruhat interval with a parabolic coset contains a
unique maximal element. As an application, we give a decomposition formula for
the Poincar\'{e} polynomial of a Coxeter group element. We also show that the
fibers of standard parabolic projection maps on Schubert varieties are
themselves Schubert varieties.
