Abstract:
In this article it is proved that over a field of characteristic zero the product V1,…,VnV1,…,Vn of varieties of Lie algebras in which VnVn is nilpotent has, as a rule, infinite base rank. An exception is the case when n=2n=2, V2V2 is abelian, and V1V1 is nilpotent. It is also shown that if V1V1 is abelian and V2=varsl2V2=varsl2, then the base rank of V1V2V1V2 is equal to two. A criterion is obtained for the finiteness of the base rank of a special variety. All special varieties of Lie algebras of almost finite base rank are described.