$n$-lie property of the Jacobian as a condition for complete integrability

We prove that an associative commutative algebra $U$ with derivations $D_1,\dots,D_n\in\operatorname{Der}U$ is an $n$-Lie algebra with respect to the $n$-multiplication $D_1\wedge\dots\wedge D_n$ n if the system $\{D_1,\dots,D_n\}$ is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. One more formulation of the Frobenius condition for complete integrability is obtained in terms of $n$-Lie multiplications. A differential system $\{D_1,\dots,D_n\}$ of rank $n$ on a manifold $M^m$ is in involution if and only if the space of smooth functions on $M$ is an $n$-Lie algebra with respect to the Jacobian $\operatorname{Det}(D_iu_j)$.