So I know that the minimum polynomial m (x) must be divided $ (x-1) (x ^ 2 + 1) ^ 2 $. As we are working with 4 for 4 real matrices,

$ m (x) = (x-1) $ or $ m (x) = x ^ 2 + 1 $ or $ m (x) = (x ^ 2 + 1) ^ 2 $ or $ m (x) = (x-1) (x ^ 2 + 1) $.

Did I just do a case-by-case analysis or is there a way to determine which of the four really is the minimum polynomial?

In addition, if it is case by case, we would have:

(I) m (x) = $ (x-1) $ corresponds to A being the 4 by 4 identity matrix

(II) $ m (x) = (x ^ 2 + 1) ^ 2 $ corresponds to the companion matrix of $ x ^ 4 + 2x ^ 2 + 1 $

(III) $ m (x) = (x ^ 2 + 1) $ means that we have two blocks each with a complementary matrix corresponding to $ x ^ 2 + 1 $

(IV) $ m (x) = (x-1) (x ^ 2 + 1) $ means that we have a block with companion matrix $ x ^ 3 – x ^ 2 + x -1 $ and another block corresponding to $ x-1 $

I'm in the correct way?