Structure of the Variety of Alternative Algebras with the Lie-Nilpotency Identity of Degree 5

We construct an additive basis for a relatively free alternative algebra of Lie-nilpotent degree 5, describe the associative center and core of this algebra, and find the T-generators of the full center. Also, we give some asymptotic estimate for the codimension of the T-ideal generated by a commutator of degree 5 in a free alternative algebra, and find a finite-dimensional superalgebra that generates the variety of alternative algebras with the Lie-nilpotency of the selfadjoint operator of degree 5.