Traditionally, Schur's lemma is stated as
Every complex matrix is unitrarily similar to an upper triangle matrix.
The theorem is big and the proof uses induction and is very lengthy.
On the other hand, the "real matrix" version of Schur's lemma states that
Every real matrix whose characteristic polynomial splits in $\Bbb R$ is orthogonally similar to an upper upper triangle matrix.
It's tempting to think that this theorem can be somehow directly deduced by the previous one without doing the induction and those heavy steps again. However, I did try to do so, but failed. So can it be done? Or is it inevitable to work through the whole induction process again to prove the "real matrix" version of Schur's lemma?