An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics

Let g g be the Lie algebra of a connected reductive group G G over an algebraically closed field of characteristic p > 0 p > 0 . Suppose that G ( 1 ) {G⁽¹⁾} is simply connected and p p is good for the root system of G G . Given a one-dimensional torus λ ⊂ G λ ⊂ G let g ( λ , 1 ) g(λ ,1) denote the weight component of Ad( λ ) {Ad(}λ {)} corresponding to weight i ∈ X ( λ ) ≅ Z i ∈ X(λ ) Z . It is proved in the paper that, for any nonzero nilpotent element e ∈ g e ∈ g , there is a one-dimentional torus λ e