I want to use the Jordan form to show that every element $A\in \mathfrak{gl}_n$ can be written as $N+S$ where $N\in\mathfrak{gl}_n$ is nilpotent and $S\in\mathfrak{gl}_n$ is semisimple and $NS=SN$. Let's put us over $\Bbb C$.
Well my first thought is, what does $A\in \mathfrak{gl}_n$ look like? Well they have the commutator as the lie bracket, so $A_1A_2-A_2A_1=0$ I believe and we need all such $A=(A_1A_2-A_2A_1)\in\mathfrak{gl}_n$ I am fairly sure.
Then what is left is to show that $A$ can have some $A_1$ as a nilpotent matrix, and $A_2$ as a semisimple matrix.
Is that what I should do?
This is just the Jordan-Chevalley decomposition of matrices in $M_n(K)$, see wikipedia. Here $K$ is assumed to be a perfect field. Of course, over the complex numbers it holds. Since $M_n(K)$ carries a Lie bracket via $[A,B]=AB-BA$, we can view $M_n(K)$ as the Lie algebra $\mathfrak{gl}_n(K)$.