I am completely stumped at the proof of Theorem 13 in Chapter 6, Hoffman and Kunze. The theorem goes:
Let $T$ be a linear operator on the finite-dimensional vector space $V$ over field $F$. Suppose that the minimal polynomial for $T$ decomposes over $F$ into a product of linear polynomials. Then there is a diagonalizable operator $D$ on $V$ and a nilpotent operator $N$ on $V$ such that
(i) $T = D + N$
(ii) $DN = ND$
The diagonalizable operator $D$ and the nilpotent operator are uniquely determined by (i) and (ii) and each of them is a polynomial in $T$.
The crux was to prove the uniqueness. In the proof given, there was a sentence: "$D'$ and $N'$ commute with any polynomial in $T$; hence they commute with $D$ and with $N$"
Does this mean $DD' = D'D$ and $NN' = N'N$? How did this come about? I am definitely overlooking something obvious...