Proof involving orthogonally diagonalizable matrix

Question

Question:

Let $A$ and $B$ be $n$ x $n$ matrices such that both A and B are orthogonally diagonalizable. Prove or disprove the following assertions:

(i) the matrix $AB$ is orthogonally diagonalizable;

(ii) the matrix $AB+BA$ is orthogonally diagonalizable.

I think I'll be using the property that $A=MTM^{-1}=MTM^T$ and same for $B$, but not really sure where to go from there? TIA.