Exercise 13, § 9

Let R be a root system in a vector space V. Two elements $v_1, v_2$ of V are said to be disjoint if R is the direct sum of two root systems $R_1$ and $R_2$ (Chap. VI, §1, no. 2) such that $v_i$ belongs to the vector subspace of V generated by $R_i,i= 1,2$. Show that two elements of $V_{\mathbf{R}}$ that belong to the same chamber of R are disjoint if and only if they are orthogonal.