Extension of the Hall-Witt identity
There is a sense in which there are no further identities. Let me explain.
In Section 5.2.3 of my paper "An infinite presentation of the Torelli group" (available on my webpage), I construct a sort of presentation of the commutator subgroup of a free group where relations are relations like the Witt-Hall relations. I actually was interested in the commutator subgroup as a subgroup of the fundamental group of a surface, so my generators are all things of the form $[x,y$] where $x$ and $y$ are simple closed curves that only intersect at the basepoint. However, I'm fairly certain you could adapt the proof there to prove the following.
Fix a nonabelian free group $F$. Let $S$ be the set of all elements of the form $[x,y]$, where $x$ and $y$ form part of a basis (any basis) for $F$. Observe that $S$ is infinite. Let $R$ be the set of all relations of the following forms. To save notation, we will denote by $[x,y]^w$ the element $[w^{-1}xw,w^{-1}yw] \in S$, where $[x,y] \in S$ and $w \in F$ is arbitrary.
$[x,y]=[y,x]^{-1}$ if $[x,y] \in S$.
$[x,y]=[z,w]$ if $[x,y] \in S$ and $[z,w] \in S$ happen to already be equal in $F$ (for example, $[yx,y] = [x,y]$).
$[z,w]^{-1} [x,y] [z,w] = [x,y]^{[z,w]}$ if $[x,y] \in S$ and $[z,w] \in S$.
$[xz,y] = [x,y]^z [z,y]$ if $x$ and $y$ and $z$ form part of a basis (any basis) for $F$.
$[x,y]^z = [z,x] [z,y]^x [x,y] [x,z]^y [y,z]$ if $x$ and $y$ and $z$ form part of a basis (any basis) for $F.
This 5th relation is a variant on the usual Jacobi identity (put into a form useful for this presentation). The conclusion is then that $[F,F]$ has the presentation with generators $S$ and relations $R$.
The point of all this is that there are no other "deeper" commutator identities that are not consequences of the ones you know.
The following article can answer your question:
https://www.researchgate.net/publication/319957676_A_generalization_of_the_Hall-Witt_identity