The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operations are:

- $a^b= b^{-1}ab$, taking the conjugate in $F$.
- $[a,b]= aba^{-1}b^{-1}$, taking the commutator of two elements in $F$.

And that's all. (If I were allowing only conjugation then this structure would be a conjugation quandle, but I'm also allowing to take commutators, but not to take products- the group product is not part of the structure). Are such structures at all studied or known?

Question: Is the full set of relations in this structure known (in the sense of universal algebra)? Is there a proof in the literature?

`$[a,b^{-1},c]^{b}[b,c^{-1},a]^{c}[c,a^{-1},b]^{a}] = 1$`

, which is reminiscent of the Jacobi identity ( I may have got the formula wrong from memory, but it can be found in group theory texts, and may be due to P. Hall). Taking the direct sum of the quotients of successive terms of the lower central series gives a Lie ring like structure which has been looked at in the literature. – Geoff Robinson Dec 4 '12 at 8:09