I was able to show that the Jordan-Chevalley decomposition of an $m \times m$ Jordan block $A$ is given by
$$A = \lambda \cdot Id_m + (A - \lambda \cdot Id_m)$$
Now I need to give a proof of the existence of the Jordan-Chevalley decomposition which does not use Jordan normal forms.
I am not even sure I understand the question. So am I just supposed to show that the Jordan-Chevalley decomposition exists for any matrix, not just Jordan blocks? Or am I supposed to prove the existence without using the concept of a "Jordan block" in my proof?
For context: this was a question on an exercise sheet provided by the university course I am taking.