  In this note, we provide a short and self-contained proof that the braid
group on n strands acts transitively on the set of reduced factorizations of a
Coxeter element in a Coxeter group of finite rank n into products of
reflections. We moreover use the same argument to also show that all
factorizations of an element in a parabolic subgroup of W lie as well in this
parabolic subgroup.

Group Theory;Combinatorics;Representation Theory