When is $\nabla\cdot({\bf a}\cdot{\bf r})({\bf b}\cdot{\bf r})\,{\bf F}({\bf r})=0$ for constant vectors ${\bf a}$ and ${\bf b}$?

Question

When is the assertion that $$\nabla\cdot({\bf a}\cdot{\bf r})({\bf b}\cdot{\bf r})\,{\bf F}({\bf r})=0$$ for constant vectors ${\bf a}$ and ${\bf b}$ and a everywhere-divergenceless ${\bf F}$ true? The reason I ask is that I wish to prove $$\iiint_VdV({\bf a}\cdot{\bf r})({\bf b}\cdot{\bf F})+({\bf b}\cdot{\bf r})({\bf a}\cdot{\bf F})=0$$ for the above vectors (with the additional requirement that ${\bf F}$ vanishes outside the enclosed volume $V$), and have obtained $({\bf a}\cdot{\bf r})({\bf b}\cdot{\bf F})+({\bf b}\cdot{\bf r})({\bf a}\cdot{\bf F})$ from the above divergence, but cannot proceed without proving that this divergence is indeed zero.