Geodesic
The base rank of varieties of Lie algebras
Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 15-31

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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. 
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     author = {M. V. Zaicev},
     title = {The base rank of varieties of {Lie} algebras},
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M. V. Zaicev. The base rank of varieties of Lie algebras. Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 15-31. http://geodesic.mathdoc.fr/item/SM_1995_80_1_a1/