It's quite simple, it takes two seconds to understand it, but I can't find a quick demonstration. Let's say I have three matrices $A$, $B$ and $C$. The matrices $A$ and $B$ do not commute with one other, but $C$ commutes with both.
Does this imply that $C$ must be proportional to the identity?
The matrices $A$ and $B$ are supposed invertible.