Suppose you have a symmetric real 3x3 Matrix $S$ and an orthogonal matrix $O$ such that $O$ commutes with $S$, i.e. $OS = SO$. Suppose that $O$ is a nontrivial rotation about an axis in direction of $n \in \mathbb{R}^3$, i.e. $On = n$ and $O \neq \mathrm{id}$.
Is it true that $n$ must be an eigenvector of $S$? If so, how to prove it?
If it is not true, are there slight modifications of the suppositions which make it true?
If it is true, is there a simple way to generalize this statement?