I've wondered for a while about the (Hall-)Witt identity in group theory:

$[[a,b^{-1}],c]^b \cdot [[b,c^{-1}],a]^c \cdot [[c,a^{-1}]],b]^a = 1$.

(Here, $x^y$ means $y^{-1}xy$ and $[x,y]$ means $x^{-1}y^{-1}xy$.) Does anybody have any motivation for this? To me, it almost seems like it comes out of nowhere so that we can prove the three subgroup lemma or something. Is there some reason to expect a relation like this to hold, or a way of reducing it to simpler relations in a meaningful way? Perhaps we should expect something like this from the free-ness of the commutator subgroup of the free group on three letters? Or should we expect some analogue of the Jacobi identity to hold, and if so, why?