The Existence of Root Subgroup Translated by a Given Element into its Opposite

Resumo

Let Φ be a simply laced root system, K an algebraically closed field, and G = G_ad(Φ,K) the adjoint group of type Φ over K. Then for every nontrivial element g ∈ G there exists a root element x of the Lie algebra of G such that x and gx are opposite.