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

We construct an additive basis for a relatively freealternative algebra of Lie-nilpotent degree 5,describe the associative center and core of this algebra, and findthe T-generators of the full center.Also, we give some asymptotic estimate for the codimensionof the T-ideal generated by a commutator of degree 5in a free alternative algebra, and finda finite-dimensional superalgebra thatgenerates the variety of alternative algebraswith the Lie-nilpotency of the selfadjoint operator of degree 5.