Here is an exercise which is meant to prove the existence of the Jordan-Chevalley decomposition of a complex matrix.
Let $A \in M_{n}(\mathbb{C})$, and let $\mu_{A} = \prod\limits_{k = 1}^{r} (X - \lambda_{k})^{n_{k}}$ be its minimal polynomial (with $\lambda_{k}$ distinct complex numbers, $n_{k} > 0$).
Show that $\{ P \in \mathbb{C}[X]/ P(A)$ is nilpotent $\} = \{ Q(X)(X - \lambda_{1}) ... (X - \lambda_{r}) / Q \in \mathbb{C}[X] \}$, and use it to prove that there exists a polynomial $P$ such that $P(A)$ is nilpotent and $A - P(A)$ is diagonalizable.
I have no issues with the first statement, but I cannot manage to find a polynomial $P$ satisfying what we are looking for ...
Thank you for your help.