## On the finite basis property of some solvable varieties of Lie algebras of finite axiomatic rank.(Russian)Zbl 0594.17008

Let $${\mathfrak N}_t$$ denote a variety of Lie algebras over a field $$k$$ defined by an identity $$x_ 0(\text{ad}\, x_1) \cdots (\text{ad}\, x_t)=0$$. This variety consists of all Lie $$k$$-algebras which are nilpotent of class at most $$t$$. Denote by $${\mathfrak N}_t {\mathfrak N}_s$$ the variety of all Lie algebras which are extensions of $${\mathfrak N}_t$$-algebras by $${\mathfrak N}_s$$-algebras.
Theorem. Let $$V$$ be a subvariety in $${\mathfrak N}_t {\mathfrak N}_s$$ defined by a set of identities depending on a finite set of variables. If $$| k| \geq t+1$$, then $$V$$ is finitely based.
In particular, if char k$$=0$$, then any subvariety in $${\mathfrak N}_t {\mathfrak N}_1$$ is finitely based [A. N. Krasil’nikov, Vestn. Mosk. Univ., Ser. I 1982, No. 2, 34–38 (1982; Zbl 0494.17010)].

### MSC:

 17B30 Solvable, nilpotent (super)algebras 17B99 Lie algebras and Lie superalgebras

Zbl 0494.17010
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