A family of simple Lie algebras in characteristic two

As reported by A.I. Kostrikin in his paper [21], the very first algebra which distinguished modular (i.e. over fields of positive characteristic p) Lie algebra theory from the classical one was the Witt algebra W (1 : k), where k is a power of p. This algebra is a generalization due to H. Zassenhaus [31] in the thirties of an analogous structure defined by E. Witt over the integers. This algebra is graded over the elementary abelian additive group of the field Fk and, for p > 2, it is simple. When the characteristic is two W (1 : k) has exactly one non-trivial ideal, namely its derived subalgebraZk. The simple object is called a Zassenhaus algebra. In characteristic two, a Zassenhaus algebra admits a non-singular outer derivation, which is quite an important feature (see [3]). In the following years, further generalizations of the above structure led to the construction of other families of simple Lie algebras,called algebras of Cartan type (see [29]), namely the generalized Jacobson-Witt algebras, the special algebras, the hamiltonian algebras and the contact algebras. Kostrikin and I.R. Shafarevich during the sixties conjectured that, apart from the small characteristic case, every finite-dimensional simple Lie algebra over an algebraically closed field is either classical or of Cartan type. Recently this was proved true for p > 7 by H. Strade