The Jordan-Chevalley-Theorem states that for a given endomorphism $f\in \mathrm{End}_K(V)$ of a $K$-vector space $V$, such that the characteristic polynomial $\chi_f$ splits into factors, there exist unique endomorphisms $f_d,f_n\in \mathrm{End}_K(V)$ sucht that $$ f= f_d+f_n~\mathrm{and}~f_d\circ f_n = f_n\circ f_d.$$
The key to proving the uniqueness is to show that for different endomorphisms $f_d',f_n'$ that satisfy these requirements, $f_n-f_n'$ is again nilpotent. This follows easily from binomials theorem, provided that $f_n\circ f_n' = f_n'\circ f_n$ holds.
However, in every text that I found, it is only stated that this is a consequence from $f\circ f_n' = f_n'\circ f_n$. But I don't see how this implies that the two nilpotent maps actually do commute. Is there a simple manipulation I am missing, or is there more to it?