  We clarify the relationship between the linear commutator and the ordinary
commutator by showing that in any variety satisfying a nontrivial idempotent
Mal'cev condition the linear commutator is definable in terms of the
centralizer relation. We derive from this that abelian algebras are
quasi-affine in such varieties. We refine this by showing that if A is an
abelian algebra and V(A) satifies an idempotent Mal'cev condition which fails
to hold in the variety of semilattices, then A is affine.
