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 S3={σ,ρ σ2=ρ3=1,σρσ=ρ2} such that for any x L we have (xσ−x)+(xσ−x)ρ+(xσ−x)ρ2=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.