The base rank of varieties of Lie algebras

In this article it is proved that over a field of characteristic zero the product $V_1,\dots,V_n$ of varieties of Lie algebras in which $V_n$ is nilpotent has, as a rule, infinite base rank. An exception is the case when $n=2$, $ V_2$ is abelian, and $V_1$ is nilpotent. It is also shown that if $V_1$ is abelian and $V_2=\operatorname{var\,sl}_2$, then the base rank of $V_1V_2$ 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.