The history of the notion is intertwined with that of a quasi-abelian category, as, for awhile, it was not known whether the two notions are distinct (see quasi-abelian category#History).
The two properties used in the definition can be characterized by several equivalent conditions.
[1] Every semi-abelian category has a maximal exact structure.
If a semi-abelian category is not quasi-abelian, then the class of all kernel-cokernel pairs does not form an exact structure.
By dividing the two conditions on the induced map in the definition, one can define left semi-abelian categories by requiring that
The same holds, if the category is right semi-abelian and left quasi-abelian.