Although this question is mostly out of curiosity (as of now), I hope it is nevertheless suitable for MO.


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[This very recent (and still open)][1] question about the [**Hall-Witt identity**][2] led me to wonder:

> 1. Does some related generalization to 4 or more variables exist?

And also, 
> 2. Does that then imply a 4 variable (or more) Jacobi like identity?


  [1]: https://mathoverflow.net/questions/81224/motivation-for-hall-witt-identity
  [2]: http://en.wikipedia.org/wiki/Commutator#Identities