  We compare the 'Smith is Huq' condition (SH) with three commutator conditions
in semi-abelian categories: first an apparently weaker condition which arose in
joint work with Bourn and turns out to be equivalent with (SH), then an
apparently equivalent condition which takes commutation of non-normal
subobjects into account and turns out to be stronger than (SH). This leads to
the even stronger condition that weighted commutators (in the sense of Gran,
Janelidze and Ursini) are independent of the chosen weight, which is known to
be false for groups but turns out to be true in any two-nilpotent semi-abelian
category.
