Let $G$ be a group and let $[a,b]=a^{-1}b^{-1}ab$ be the commutator of $a$ and $b$ in $G$. There are several well-known commutator identities such as

$[x, z y] = [x, y]\cdot [x, z]^y$

and

$[[x, y^{-1}], z]^y\cdot[[y, z^{-1}], x]^z\cdot[[z, x^{-1}], y]^x = 1$.

I wonder if there is a more complete list of commutator identities and commutator equivalences of the form

$\alpha(x_1,\cdots,x_n)\equiv\beta(x_1,\cdots,x_n)\mod{N}$,

where $\alpha$ and $\beta$ are expressions involving commutators and $N$ is some subgroup of $G$, for example the $k$-th term in the lower central series of $G$. Anyone knows such a list? Thank you!