Homework 8. (Due Apr.19)

Let $E,(,),R$ be a root system with set of simple roots \Pi.
Let s be an isomorphism of E with itself such that (sx,sy)=(x,y) for all
x,y in E, s(\Pi)=\Pi.

(i) Show that s(R)=R.
(Hint. Use that any root is of the form r_{a_1}r_{a_2}...r_{a_n}a' where 
a_1,a_2,...,a_n,a' are simple roots and for a in\Pi, r_a is the reflection 
with respect to a.  Use that sr_as^{-1}=r_{s(a)} for a in\Pi.)

(ii) Let E_s={e in E; s(e)=e} with the symmetric bilinear form induced by (,).
 For each orbit \o of s on \Pi let
a_\o=\sum_{a\in\o}a; assume that if a,a'\in\o are distinct then (a,a')=0.

Let \Pi_s be the set of vectors a_\o for various \o as
above. Show that \Pi_s is the set of simple roots of a root system in E_s
of type

G_2 if R is of type D_4 and s^3=1,s not 1
F_4 if R is of type E_6 and s^2=1, s not 1.