Exercise 8, § 1

Construct a group W and two subsets S and S' of W such that (W, S) and (W, S') are non-isomorphic Coxeter systems, one of them being irreducible and the other not (for W take a dihedral group of order 12 generated by {s, s'} where s and s' are of order 2 and s ≠ s', and put S = {s, s'} and S' = {(ss')^3, s', s'(ss')^2}).