Commutators in semi-simple algebraic groups

Notation and preliminary. We shall use the terminology and results in 1. Let G be a connected semi-simple algebraic group defined over an algebraically closed field K, T a maximal torus of G, B a Borel subgroup containing T, Bu the unipotent part of B, N the normalizer of T in G, W=N/T the Weyl group, and X=X(T) the character group of T. The character group X is a free abelian group of finite rank n. The Weyl group W acts on X by