N‐commutators for simple Lie algebras

The N‐commutator $sN(X1,…,XN)$ of N vector fields $X1,…,XN$ (differential operators of order 1) is defined as the skew‐symmetric sum of the N! compositions $Xσ(1)⋯Xσ(N)$ for all permutations $σ∈SymN.$ In general it is a differential operator of order more than 1, but for some special cases of N it might happen that $sN$ will be a well‐defined operation on the space of vector fields. We construct such N‐commutators for simple Lie algebras for the four classical series and for the exceptional algebra $G2.$ We establish that $G2$ has well‐defined 2‐ and 10‐commutators.