How to realize Lie algebras by vector fields

Abstract

An algorithm for embedding finite dimensional Lie algebras into Lie algebras of vector fields (and Lie superalgebras into Lie superalgebras of vector fields) is offered in a way applicable over ground fields of any characteristic. The algorithm is illustrated by reproducing Cartan's interpretations of the Lie algebra of G(2) as the Lie algebra that preserves certain non-integrable distributions. Similar algorithm and interpretation are applicable to other exceptional simple Lie algebras, as well as to all non-exceptional simple ones and many non-simple ones, and to many Lie superalgebras.