When is B-A- a generalized inverse of AB ?

In practice factorizations of a generalized inverse often arise from factorizations of the matrix which is to be inverted. In addition to full rank factorizations, normal factorizations and singular value decompositions (SVD) there are other factorizations of particular matrices that are natural to certain problems, for instance in statistics. Answers to questions like "when is B-A- a generalized inverse of AB?" may thus be a computational tool. Besides they are of significant interest because they provide us with insights into the g-inversion of matrix products; compare [9].