On Engel words on simple algebraic groups

Abstract

Let G be a simple algebraic group defined over an algebraically closed field K and let G=G(K). We consider here the problem when an element g∈G can be presented in the form g=en(g1,g2):=[g1,[g1,⋯[g1︸n-times,g2]⋯]] for some g1,g2∈G. This can always be done if g is a semisimple or a unipotent element. Also, it is known that this holds for G=SL2(K). Here we prove that g=en(g1,g2) for every n if G is a group of type B2 or G2 and we prove g=e2(g1,g2)=[g1,[g1,g2]] if G=PGL3(K). We also show g=e2(g1,g2)=[g1,[g1,g2]] if g is a regular element of G and G is a group of rank 3. For any simple group G we give a criterion for some “general” regular elements of G to be presented in the form [g1,[g1,g2]].