I was wondering if there's any attempt to generalize the commutator for something general for more than two terms.


Here's what I was thinking of, for $[A,B]=AB-BA$, so for three terms:


$[A,B,C] = ABC+BCA+CAB-BAC-CBA-ACB$


where I have taken a plus sign for a cyclic permutation of $ABC$, and a minus sign for acyclic permutation of $ABC$.


Is such a generalization still called commutator or something else?



Generalizing the commutator and anti-commutator
This generalization is called a ternary commutator, and there are $n$-commutator generalizations, as explained [here:](http://arxiv.org/abs/1207.6312)


![](https://ilorentz.org/beenakker/MO/n_commutators.png)


