Exercise 14, § 2

Let $(G, B, N, S)$ be a Tits system and $(I, \mathscr{U})$ the associated numbered structured building (Exerc. 12). Show that G, considered as a group of automorphisms of I, satisfies the conditions of Exerc. 13. With the notation of Exerc. 12, put $A = \varphi_e(A_0)$ and $C = \varphi_e(C_e)$; show that B is the stabilizer of C in G. Let $\tilde{N}$ be the stabilizer of A in G; show that $\tilde{N}/(B \cap \tilde{N})$ can be identified with W and that $(G, B, \tilde{N}, S)$ is the saturated Tits system associated to $(G, B, N, S)$ (Exerc. 5).