# Is the theorem of complete reducibility or the abstract Jordan decomposition needed for structure theory of semisimple Lie algebras?

Let $\mathfrak g$ be a complex semisimple Lie algebra. A theorem of Weyl says that every finite-dimensional representation of $\mathfrak g$ is completely reducible. Another very important theorem is that the common Jordan decomposition of endomorphisms in a complex vector space has an abstract counterpart in abstract semisimple Lie algebras $\mathfrak g$. My question is basically: do we need these two theorems for proving the basic facts about the root space decomposition (i.e. structure theory of semisimple Lie algebras)?

1) One example: in many places, the proof that a Cartan subalgebra $\mathfrak h$ (defined generally as nilpotent algebra that coincides with its normalizer in $\mathfrak g$) of a semisimple Lie algebra consists entirely of elements $X$ such that $ad(X)$ is semisimple - is based on above mentioned abstract Jordan decomposition. But in Dixmier, Enveloping algebras, page 38, Dixmier does not seem to use the abstract Jordan decomposition but only the usual one for endomorphisms (and the weight decomposition of $\mathfrak g$ given by the adjoint representation of $\mathfrak h$ on $\mathfrak g$ and that all derivations of a semisimple Lie algebra are inner).

2) The proof of the abstract Jordan decomposition is in many places (e.g. Humphreys) based on the theorem of Weyl about complete reducibility. But in Kirillov, An Introduction to Lie groups and Lie algebras, chapter 6.4, the proof does not seem to be needing Weyl's theorem.

3) A remark: I know that the fact that all derivations of $\mathfrak g$ are inner can be proved with the theorem of Weyl. But it can also be proved using 'only' the nondegeneracy of the Killing form. So here again we do not need the theorem of Weyl.

4) So summarizing I do not see where we would need these two theorems to show that a Cartan subalgebra of $\mathfrak g$ is abelian and consists of semisimple elements. But given that, do we need at any point of proving the basic properties of the root space decomposition these two theorems? In my understanding, only the classification of irreducible $sl_2$-modules plays a role here. We do not need to know that any finite-dimensional module is a direct sum of irreducible ones or that $\rho(H)$ for $H \in \mathfrak g$ semisimple, is semisimple for any representation $\rho$.

I will try an answer. Let us start with the title question. Yes, the abstract Jordan decomposition is used in the theory of semsimple Lie algebras, see this MSE-question. Weyl's theorem itself seems more important for the representation theory of semisimple Lie algebras. But remark $3$ is true. We could prove that a semisimple Lie algebra $L$ of characteristic zero is complete, i.e., that $Der(L)=ad(L)$ and $Z(L)=0$ by Weyl's theorem. Algebraically, however, we would probably replace Weyl's theorem with Whitehead's first Lemma, that $H^1(L,M)=0$ for all finite-dimensional $L$-modules $M$, and $L$ semisimple. For the adjoint module $M=L$ we obtain then $Der(L)=ad(L)$.