Definition of N-commutator Warnings:

*

*I am not a mathematician and
I am asking this question
out of curiosity.


*This question might not have
the correct tags since I'm
not sure about what this
question
is about.
Background:
So I have seen the definition of a commutator which is
$ [A,B]=AB-BA$.
I started wondering that is there a definition for $[A,B,C]$? So I started searching and I found the definition for ternary commutator
(https://en.m.wikipedia.org/wiki/Ternary_commutator)
In the wikipedia page there is a reference to N-commutator but there is no wikipedia page for it.
I am not sure that how the extension from 2-commutator (original commutator) to 3-commutator (ternary commutator) works. So I can't figure out the definition for N-commutator.
Question:
So, I wanna know that how the extension works and what is the definition of N-commutator.
 A: The ternary commutator for an associative binary operator adds together products from all $3!$ permutations of $a,\,b,\,c$, but with a coefficient of $1$ ($-1$) for the even (odd) permutations $abc,\,bca,\,cab$ ($acb,\,bac,\,cba$). The $2$-commutator is the same basic idea, although in that case we'd deem such a description "overkill": the even (odd) permutation $ab$ ($ba$) gets a coefficient $1$ ($-1$). The general $N$-commutator is$$[a_1,\,\cdots,\,a_N]=\sum_{\sigma}\varepsilon_\sigma\prod_{i=1}^Na_{\sigma_i},$$where:

*

*The sum is over all permutations $\sigma$ of $1,\,\cdots,\,N$;

*The Levi-Civita symbol $\varepsilon_\sigma$ is $1$ ($-1$) for even (odd) $\sigma$;

*The product is left-to-right, e.g. if $\sigma$ is the identity operator $\prod_{i=1}^Na_{\sigma_i}=a_1a_2\cdots a_n$.

The $N$-commutator is multilinear, fully antisymmetric and invariant under cyclic permutations, and satisfies$$[a_1,\,\cdots,\,a_N]=\sum_{i=1}^Na_i[a_{i+1},\,\cdots,\,a_N,\,a_1,\,\cdots,\,a_{i-1}],$$e.g. $[a,\,b,\,c]=a(bc-cb)+b(ca-ac)+c(ab-ba)$.Further,$$[a_{\sigma_1},\,\cdots,\,a_{\sigma_N}]=\varepsilon_\sigma[a_1,\,\cdots,\,a_N].$$
