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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 \(L_2\cong A\otimes_K {sl}_2\).
17B01 Identities, free Lie (super)algebras
17B05 Structure theory for Lie algebras and superalgebras