Commutator Identity and Commutator Subgroup
B. H. Neumann proved that there is a $4$-generator, non-metabelian group, all of whose $3$-generated subgroups are metabelian. So in this group the identity $$\bigl[[a,b],c\bigr]\cdot\bigl[[b,c],a\bigr]\cdot\bigl[[c,a],b\bigr]=1$$ holds.
Neumann' s example is a finite $2$-group. Compare with Theorem 4 of this paper by Alperin.