You might be right. I wasn't sure and I didn't need to assume it, so I didn't.
No, I said "of the form described" - i.e. the 3-block decomposition they were claiming exists. The point being the only way it doesn't exist is if there are only 2 blocks corresponding to powers of f.No. E.g. if I is the 2x2 identity matrix then it's minimal polynomial is (x-1) (and so (x-1) only divides it once) but it has 2 blocks in its decomposition.