A note on the transitive Hurwitz action on decompositions of parabolic   Coxeter elements
  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. 
Barbara Baumeister and Matthew Dyer and Christian Stump and Patrick   Wegener
math.GR math.CO math.RT