  We study the relation between Bourn's notion of peri-abelian category and
conditions involving the coincidence of the Smith, Huq and Higgins commutators.
In particular we show that a semi-abelian category is peri-abelian if and only
if for each normal subobject $K\leq X$, the Higgins commutator of $K$ with
itself coincides with the normalisation of the Smith commutator of the
denormalisation of $K$ with itself. We show that if a category is peri-abelian,
then the condition (UCE), which was introduced and studied by Casas and the
second author, holds for that category. In addition we show, using amongst
other things a result by Cigoli, that all categories of interest in the sense
of Orzech are peri-abelian and therefore satisfy the condition (UCE).
