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:

1. $a^b= b^{-1}ab$, taking the conjugate in $F$.
2. $[a,b]= aba^{-1}b^{-1}$, 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?
3 make the elements all elements of F, so the question of "all relations" makes more sense.

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:

1. $a^b= b^{-1}ab$, taking the conjugate in $F$.
2. $[a,b]= aba^{-1}b^{-1}$, 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). I want the structure to be closed under these two binary operations. Are such structures at all studied or known?

Question: Is the full set of relations in this structure known? Is there a proof in the literature?
2 added 69 characters in body

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

1. $a^b= b^{-1}ab$, taking the conjugate in $F$.
2. $[a,b]= aba^{-1}b^{-1}$ 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). I want the structure to be closed under these two binary operations. Are such structures at all studied or known?

Question: Is the full set of relations in this structure known? Is there a proof in the literature?
1