So I have a question about the additive Jordan decomposition in Springer's book on linear algebraic groups. If we have a morphism $f:V\Rightarrow W$ with $V,W$ vector spaces and $a\in End(V), b \in End(W)$, then we have the following statement:
$$f\circ a=b\circ f\Rightarrow f\circ a_n=b_n\circ f,f\circ a_s=b_s\circ f$$
Here the index n indicates the nilpotent component and the index s the diagonalisable component. I was able to reduce this question to the case where f is either surjective or injective, but I do not know how to continue. Any help is appreciated.
