The Engel identity and its application

In the case of characteristic zero, the Engel identity implies nilpotence in the variety generated by simple infinite-dimensional Lie algebras of Cartan type. An analogous result is also true for 2-metabelian Lie algebras (an algebra is called 2-metabelian if every 2-generator subalgebra is metabelian) over a field whose characteristic does not divide $5!\,$, which in this case permits one to prove solvability of the variety of 2-metabelian Lie algebras.
Bibliography: 10 titles.