I was reading a paper, and it made a claim that for some nilpotent matrix $A$, we can say that we can find a Jordan basis of $A$ that is orthonormal. I understand that what this means is that all sets of Jordan block subspaces are mutually orthogonal (I call it this for lack of a better name). I can show that given a Jordan block subspace, the Jordan basis can be orthogonal, but I am struggling to show that each of the subspaces are mutually orthogonal.
Is this a true statement, or are there more assumptions that must be being made that I didn't see?