Lie algebras with triality

By analogy with the definition of group with triality, we introduce Lie algebra with triality as Lie algebra L which admits the group of automorphisms S-3 = {sigma, rho \ sigma(2) = rho(3) = 1, sigmarhosigma = rho(2)} such that for any x is an element of L we have (x(sigma) - x) + (x(sigma) - x)(rho) + (x(sigma) x)(rho2) = 0. We describe the structure of finite-dimensional Lie algebra with triality over a field of characteristic 0 and give applications of Lie algebras with triality, to the theory of Malcev algebras. (C) 2003 Elsevier Inc. All rights reserved.