Apologies in advance if my formatting is bad.
I'm working on the following problem and I've run into a bit of a wall.
Let $ V $ be a vector space over an algebraically closed field $ \mathcal{F} $. Let $ T \in End(V) $ with Jordan decomposition $ T = D + N $ where $ D $ is diagonalizable and $ N $ is nilpotent. Prove that $ S \in End(V) $ commutes with $ T $ if and only if $ S $ commutes with $ D $ and $ S $ commutes with $ N $.
Proving the reverse direction is very easy. If $ S $ commutes with $ D $ and $ N $, then
$$ ST = S(D + N) = SD + SN = DS + NS = (D + N)S = TS $$
So $ T $ and $ S $ commute.
Proving the forward direction is more difficult. If $ T $ and $ S $ commute, then
$$ ST = TS \Rightarrow SD - DS = NS - SN $$
At this point, I'm not sure how to proceed. I've tried messing around with the minimal polynomial of $ T $ but was unable to make something of it. I've also tried applying the Jordan decomposition to $ S $ itself, but this didn't help either. I feel like I'm missing some fundamental property about diagonalizable and nilpotent operators that I need to complete this proof. Any suggestions or pointers on how to proceed from here would be greatly appreciated.