  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 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. We introduce a notion of Jacobi element of the group ring
$\mathbb{Z}[S_n]$ and describe them on the language of equations on
coefficients. Using this description we obtain a purely combinatorial necessary
and sufficient condition for a subset to be Jacobi.
