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It is easy to check that the identity $[x_1, x_2, x_3, x_4] + [x_2, x_1, x_4, x_3] + [x_3, x_4, x_1, x_2] + [x_4, x_3, x_2, x_1] = 0$ holds in any Lie algebra as well. I. Alekseev in his recent work introduced the notion of Jacobi subset of the symmetric group $S_n$. It is a subset of $S_n$ that gives an identity of this kind. 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