Finite dinilpotent groups of small derived length

A finite dinilpotent group G is one that can be written as the product of two finite nilpotent groups, A and B say. A finite dinilpotent group is always soluble. If A is abelian and B is metabelian, with |A| and |B| coprime, we show that a bound on the derived length given by Kazarin can be improved. We show that G has derived length at most 3 unless G contains a section with a well defined structure; in particular if G is of odd order, G has derived length at most 3.