Square matrices such that $A = B + C$ with each matrix diagonalisable. Must $B$ and $C$ commute?

Question

Suppose $A, B, C$ are square, real, symmetric, positive semi-definite matrices such that $$ A = B + C $$ and $A$ is positive definite.

Must $B$ and $C$ be simultaneously diagonalisable?

I'm thinking no, but can't find a counter example.

Answer 1

I think $$ B = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}, \quad C = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix} $$ give a counter-example.