The following fact is basic in the theory of complex Lie algebras:

Theorem. Let ${\mathfrak g} \subset {\mathfrak gl}_n({\bf C})$ be a simple Lie algebra, and let $x \in {\mathfrak g}$. Let $x = x_s + x_n$ be the Jordan decomposition of $x$, thus $x_s, x_n \in {\mathfrak gl}_n({\bf C})$ are commuting elements that are semisimple and nilpotent respectively. Then $x_s,x_n$ also lie in ${\mathfrak g}$.

The standard proof of this theorem (e.g. Proposition C.17 of Fulton-Harris) proceeds, roughly speaking, by using complete reducibility of ${\mathfrak g}$-modules to reduce to the case when ${\bf C}^n$ is irreducible, at which point Schur's lemma may be applied. This fact allows one to show that the abstract Jordan decomposition for simple (or semisimple) Lie algebras is preserved under linear representations.

My (somewhat informal) question is whether the above theorem can be established without appeal to the complete reducibility of ${\mathfrak g}$-modules. I would also like to exclude any use of Casimir elements or the Weyl unitarian trick (since these give completely reducibility fairly quickly), and also wish to avoid using deeper structural facts about Lie algebras, such as the theory of Cartan subalgebras or root systems. I am happy to use the Killing form and to assume that it is non-degenerate on simple Lie algebras (and more generally, to assume Cartan's criteria for solvability or semisimplicity); I am also happy to use the closely related fact that all derivations on a simple Lie algebra are inner.

The best I can do without complete reducibility is to show that $x_s = x'_s + h$, $x_n = x'_n - h$, where $x'_s \in {\mathfrak g}$ is ad-semisimple, $x'_n \in {\mathfrak g}$ is nilpotent, and $h \in {\mathfrak gl}_n({\bf C})$ is nilpotent and commutes with every element of ${\mathfrak g}$; one can also show that $h$ vanishes when restricted to ${\mathfrak gl}(W)$ for every ${\mathfrak g}$-irreducible module $W$ of ${\bf C}^n$ (this is basically a Schur's lemma argument after observing that $h$ has trace zero on $W$), which shows that $h$ vanishes if one assumes complete reducibility. But it seems remarkably difficult to conclude the argument without complete reducibility.

Alternatively, I would be happy to see a proof of complete reducibility that did not use any of the other ingredients listed above (Casimirs, the unitary trick, or root systems). (My motivation for this is that I have been trying to arrange the foundational theory of finite-dimensional complex Lie algebras in a way that moves all the "elementary" theory to the front and the "advanced" structural theory to the back, at least according to my own subjective impressions of the elementary/advanced distinction. I had thought I had achieved this to my satisfaction in these notes, until someone recently pointed out a gap in my proof of the above theorem, which I have thus far been unable to bridge without using complete reducibility.)