The alt-comm repository contains a note giving an alternate decription of the (non-modular) commutator that yields a polynomial time algorithm for computing it. This is inspired by the alternate description of the commutator given by Kearnes in [1].

[1] Keith Kearnes.
Varieties with a difference term.
*J. Algebra*, 177(3):926–960, 1995.

[[1]]: http://dx.doi.org/10.1006/jabr.1995.1334