Exercise 20, § 4

Assume that the group G is almost simple; let H be a principal subgroup of G, of rank $\geq 2$.

a) Show that H is semi-simple of type $B_h,C_h,F_4$ or $G_2$ (use Exerc. 19).

b) Assume that H is of rank $\geq 3$. Show that G is of type $A_l,D_l,E_6,E_7$ or $E_8$ (consider the terminal vertices of the Dynkin graph of $R(G,T)$, and apply Exerc. 19).

c) Show that, if rk $H\geq 3$, we are in one of the following situations:

G is of type $A_{2l}(l\geq 3)$ and H of type $B_l$;

G is of type $A_{2l-1}(l\geq 3)$ and H of type $C_l$;

G is of type $D_l(l\geq 4)$ and H of type $B_{l-1}$;

G is of type $E_6$ and H of type $F_4$.

d) Show that, if H is of type $B_2$, then G is of type $A_3$ or $A_4$.

e) Show that, if H is of type $G_2$, then G is of type $B_3,D_4$ or $A_6$.

(For a more explicit description of these situations, see §5, Exerc. 5.)

f) Let K be a connected closed subgroup of G containing H; show that either K = G or K = H, or H is of type $G_2, K$ of type $B_3$ and G of type $D_4$ or $A_6$.