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By definition the identities $[x_1, x_2] + [x_2, x_1] = 0$ and $[x_1, x_2,x_3] + [x_2, x_3, x_1] + [x_3, x_1, x_2] = 0$ hold in any Lie algebra. It iseasy 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 aswell. I. Alekseev in his recent work introduced the notion of Jacobi subset ofthe symmetric group $S_n$. It is a subset of $S_n$ that gives an identity ofthis kind. We introduce a notion of Jacobi element of the group ring$\mathbb{Z}[S_n]$ and describe them on the language of equations oncoefficients. Using this description we obtain a purely combinatorial necessaryand sufficient condition for a subset to be Jacobi.

Generalized Jacobi identities and Jacobi elements of the group ring of the symmetric group