lower bound of norm

Question

Is there any analytic form for a lower bound of $\|AB-BC\|_F$ ? $C$ is a similarity transform of $A$, i.e., $C = P^{-1} A P$. I know that $\|AB\|_F$ is bounded by $\|A^{-1}\|_2^{-1} \|B\|_F$ and $\|A\|_F \|B\|_F$ and $\|BC\|_F$ is bounded by $\|C^{-1}\|_2^{-1} \|B\|_F$ and $\|C\|_F \|B\|_F$ Intuitively the lower bound for $\| AB - BC \|_F$ is linear to $\| B \|_F$, but I cannot prove it.