10-Commutators, 13-commutators and odd derivations

The anti-symmetric sum S N (X 1 ,..., X N ) of N! compositions of N vector fields X 1 ,..., X N ∈ V ect(n) in all possible order is said to be a N-commutator if S N (X 1 ,..., X N ) ∈ Vect(n) for any X 1 ,...,X N ∈ Vect(n) and does not vanish for some vector fields X 1 ,...,X N . We construct 10- and 13-commutators on Vect 0 (3) and 10-commutator on the space of divergence-free vector fields Vect 0 (3). We show that there are no other N-commutators on Vect(3) except for 2-, 10- and 13-commutators, and no other N-commutators on the Lie algebra of divergence-free vector fields V ect 0 (3) except for 2-, 10-commutators. These constructions are based on calculation of powers of odd derivations.