Exercise 7, § 4

Let R be an irreducible reduced root system.

a) Verify that the canonical homomorphism from $A(R)/W(R)$ to the group of automorphisms of $P(R)/Q(R)$ is injective.

b) Deduce that $-1$ belongs to $W(R)$ if and only if $Q(R) \supseteq 2P(R)$.