  We provide a necessary and sufficient condition on an element of a finite
Coxeter group to ensure the transitivity of the Hurwitz action on its set of
reduced decompositions into products of reflections. We show that this action
is transitive if and only if the element is a parabolic quasi-Coxeter element,
that is, if and only if it has a reduced decomposition into a product of
reflections that generate a parabolic subgroup.
