Exercise 11, § 1

Let (W, S) be a Coxeter system. Let Γ_∞ be the graph whose set of vertices is S, two vertices s, s' being joined by an edge if and only if m(s, s') ≠ ∞. Let S_α be the connected components of Γ_∞. Show that W can be identified with the free product of the W_{S_α}. In particular, every w ∈ W can be written uniquely as a product w_1 ... w_h with w_i ≠ 1, w_i belonging to W_{S_{α_i}}, and α_i ≠ α_{i+1} for 1 ≤ i ≤ h - 1; show that the length of w is the sum of the lengths of the w_i.*