  We establish a characterization of supernilpotent Mal'cev algebras which
generalizes the affine structure of abelian Mal'cev algebras and the recent
characterization of 3-supernilpotent Mal'cev algebras. We then show that for
varieties in which the two-generated free algebra is finite: (1) neutrality of
the higher commutators is equivalent to congruence meet-semidistributivity, and
(2) the class of varieties which interpret a Mal'cev term in every
supernilpotent algebra is equivalent to the existence of a weak difference
term. We then establish properties of the higher commutator in the
aforementioned second class of varieties.
