Basis for 2x2 diagonalizable matrices?

$\textbf{Question:}$ Find a basis for the vector space of all 2x2 matrices that commute with $\begin{bmatrix}3&2\\4&1\end{bmatrix}$, which is the matrix $B$. You must find two ways of completing this problem for full credit.

$\textbf{My Attempt:}$ I found that $B$ is diagonalizable, and so any other diagonalizable 2x2 matrix $A$ will satisfy $AB=BA$. However, I cannot think of a way to form a basis for all 2x2 diagonalizable matrices. I tried to start with a diagonal matrix with distinct entries on its diagonal, but ended up running into a lot of dead ends.

Does anyone else have any ideas on how I might find this basis? Does anyone have any other potential methods of solving this problem?