Exercise 5, § 4

Let R be a reduced and irreducible root system.

a) Show that the set of roots in R of greatest length is a symmetric closed subset of R, stable under W(R). b) Show that every non-empty subset P of R that is closed, symmetric and stable under W(R) is equal to R or to the set of roots of greatest length in R.

c) Assume that $P\not= R$. Show that the root system P is of type $D_l$ (resp. $(A_1)^l,D_4,A_2$) if R is of type $B_l$ (resp. $C_l,F_4,G_2$).