2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208228Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are irreducible if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of irreducible words.Combinatorics05E15text