Exercise 21, § 4

a) Let H be a connected closed subgroup of G of rank 1. Show that the following conditions are equivalent:

(i) H is principal;

(ii) H is clean and contains a regular element of G;

(iii) there exists a principal $\mathfrak{s}\mathfrak{l}_2$-triplet $(x, h, y)$ in $\mathfrak{g}_{\mathbf{C}}$ (Chap. VIII, §11, no. 4) such that

b) Show that G contains a principal subgroup of rank 1, and that any two such subgroups are conjugate (with the notations of Exerc. 17, remark that $S =\Delta$ ).

c) Show that a connected closed subgroup of G is principal if and only if it contains a principal subgroup of G of rank 1 (use Exerc. $18a$)$)$.