On decomposability of commutators of varieties of Lie algebras and groups into products

Several cases are presented in which the commutator of two varieties of Lie algebras cannot be decomposed into a product. An example is constructed which shows that in the general case the commutator of two varieties of Lie algebras can turn out to be decomposable even if the given varieties do not have a common right-hand factor. This example can be carried over with appropriate modifications to varieties of groups.
Bibliography: 12 titles.