In many linear algebra books, the following rank inequalities are found:
Frobenius inequality Let $A$, $B$ and $C$ be three matrices such that the product $ABC$ is defined. Then $$\operatorname{rk}(ABC) + \operatorname{rk}(B) \geq \operatorname{rk}(AB) + \operatorname{rk}(BC).$$
In the special case case $B = I$, the Frobenius inequality reduces to the
Sylvester inequality Let $A$ and $B$ be two matrices such that the product $AB$ is defined. Then $$\operatorname{rk}(A) + \operatorname{rk}(B) - n \leq \operatorname{rk}(AB).$$
Now I wonder about the equality cases in those inequalities. It is common knowledge that
In the Sylvester inequality, equality holds if and only if $$\ker(A) \subseteq \operatorname{Im}(B).$$
But I didn't find anything on the Frobenius inequality. So my question is:
How can the equality case in the Frobenius inequality be characterized?