Finding all subspaces invariant under F.(2)

Question

I have a question regarding the question in the link here:

Finding all subspaces invariant under F.

The answer is so convincing in the above link but I just want to know why the matrix corresponding to the operator $\alpha$ will be diagonal, is there a theorem saying this?

EDIT:

Also I am wondering how is the answer mentioned in the above link by @Berci leads to the answer mentioned in the OP question in the above link (The answer at the back of the book)

Thanks!