Exercise 18, § 4

We retain the notations of Exerc. 16 and 17. The subgroup H is said to be principal if it is clean and if the sub-torus $\Delta$ contains a regular element (in other words, if $k= 0$).

a) Let K be a connected closed subgroup of H. Show that K is a principal subgroup of G if and only if K is principal in H and H is principal in G (use Exerc. $15c$) and $d))$.

b) Assume from now on that H is principal. Show that the union of the $R(\mu)$, for $\mu\in B_H$, is a basis of $R(G,T)$.

c) Let $\alpha \in R(G,T)$. Prove that the restriction of $\alpha$ to S is a non-zero integer multiple of a root $\lambda \in R(H,S)$; the root $\alpha$ is a sum of elements of $R(\lambda )$ (show that the intersection of L(S) with the hyperplane $\delta (\alpha ) = 0$ is a wall of L(S)). d) Let $\lambda \in R(H,S)$; show that the nodal vector $K_{\lambda}$ is a linear combination with non-negative integer coefficients of the $K_{\alpha}$ for $\alpha \in R(\lambda )$ (cf. Exerc. 16 $d$) and Chap. V, §3, no. 5, Lemma 6).