Binary lie algebras of small dimensions

We give a simpler and shorter proof of the Gainov theorem in [1], which dealt with classifying non-Lie binary Lie algebras of dimension ≠ 4 over a field of characteristic ≠ 2. Concurrently, the case of characteristic 2 is treated, and we find out an exotic 4-dimensional non-Lie Mal'tsev algebra, which is a split extension of an irreducible 1-dimensional Mal'tsev module over a simple 3-dimensional Lie algebra. © 1998 Plenum Publishing Corporation.