A short proof that if two $2\times 2$ matrices commute, then they are simultaneously diagonalizable

Question

Let $A$ and $B$ be $2\times 2$ matrices such that $AB=BA$. Is there a short proof that they are simultaneously diagonalizable? I'm aware to the usual proof of simultaneously diagonalizability in general, but can this proof be shortened if we assume that the matrices are $2\times2$?

Answer 1

Here is a counterexample. Let $$ A=B=\begin{pmatrix} 0 & 1 \cr 0 & 0 \end{pmatrix}. $$ Then $AB=BA=0$, but they are not simultaneously diagonalisable.