On simple Lie algebras over a field of characteristic 2

On the classification of simple Lie algebras of dimension seven over fields of characteristic 2

“…We hope to prove that those algebras are the unique simple 7-dimensional Lie algebras over the field k. In this paper we prove that any simple 7-dimensional Lie algebras over k of absolute toral rank 3 is isomorphic either to W 1 or H 2 . Observe that in the case of absolute toral rank 2 this fact was proved in [2]. This paper is the second part of paper [1] about simple 7-dimensional Lie algebras over a field of characteristic two.…”

“…III. If f 0 e 2 = 0, then [f 2 , e [2] 2 ] = [f 2 , t 1 + t 3 ] = f 2 = 0, again a contradiction. ◻ Moreover, L ≃ H 2 if and only if L contains a simple 3-dimensional 2-subalgebra.…”

“…Let T = k{e [2 n ] i |i = 1, 2, 3;n = 1, 2, … } . Then T is an abelian 2-subalgebra of End k V. Since [e i , e [2] j ] = e i , i ≠ j, we have…”

“…hence ( −1 (1 + ) + ) = 1 + + = 1 or (1 + ) = 0 or = 1. [2] 1 = t 2 + t 3 , e [2] 2 = t 1 + t 3 , e [2] 3 = t 1 + t 2 .…”

“…We know two simple 7-dimensional Lie algebras over k, the Cartan algebra W 1 and the Hamilton algebra H 2 . Recall the definition of those algebras from [2]. In the two following tables we give the multiplication of the 2-closure W of the Witt-Zassenhaus algebra W 1 and the 2-closure H of the Hamilton algebra H 2 .…”