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On perfect finite-dimensional Lie algebras, satisfying the standard Lie identity of degree 5. (Russian. English summary) Zbl 0933.17003
Suppose that $$L$$ is a finite-dimensional Lie algebra over a field $$K$$ of zero characteristic. By Levi’s theorem it is a semidirect sum of a solvable radical $$L_1$$ and a semisimple Lie algebra $$L_2$$. The authors specify the structure of such algebra $$L$$ that satisfies the standard Lie identity of degree 5. They prove that $$L=L_1\oplus L_2$$, $$L_2\cong A\otimes_K {sl}_2$$, where $$A$$ is some commutative associative algebra with unit. A Lie algebra is said perfect if it coincides with its commutator subalgebra. Moreover, it is proved that any perfect Lie algebra $$L_2$$ with standard identity of degree 5 has a decomposition $$L_2\cong A\otimes_K {sl}_2$$.
##### MSC:
 17B01 Identities, free Lie (super)algebras 17B05 Structure theory for Lie algebras and superalgebras