AN EXTENSION OF A RESULT BY STEINBERG TOWARDS A CHEVALLEY BASIS ALGORITHM

Abstract. Let L be the Lie algebra of a simple algebraic group defined over F and let H be a split Cartan subalgebra of L. Let R = (X,Φ, Y,Φ∨) be the root datum of L, so that H = Y ⊗ F, and let 〈·, · 〉 : Φ × Φ ∨ 7 → Z be the corresponding bilinear form. This bilinear form induces a linear form on the roots of L by defining α: h 7→Pi〈α, yi〉ti, where h =Pi yi ⊗ ti. Given a root α, we define the multiplicity of α in L to be the number of β ∈ Φ such that α = β. For R of adjoint type, Steinberg gave an overview of the cases where mul-tiplicities greater than 1 occur. In this paper we give a complete overview of these cases, for R of any isogeny type. 1