Here is what I mean by "Cartan's semisimplicity criterion":


Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Assume that the center of $\mathfrak g$ is trivial. Then, the following three assertions are equivalent:


(1) The Killing form on $\mathfrak g\times \mathfrak g$ is nondegenerate.


(2) Every short exact sequence of finite-dimensional representations of $\mathfrak g$ splits.


(3) Every subrepresentation of the adjoint representation of $\mathfrak g$ has a complementary subrepresentation.


What I am looking for is a slick proof for this equivalence (although the only thing I really need is a proof of (1) $\Longrightarrow$ (3)). I am aware of the proof in Fulton-Harris Appendix C, but this could fill an hour of talking and seems to involve many unmotivated ideas. Is there something more explanatory? Using cohomology perhaps? Is the whole thing obvious from an advanced viewpoint? (I don't mean using the classification of simple Lie algebras, of course...) Maybe newer ideas such as Lie algebroids, algebraic groups etc. can help?



Can Lie algebra cohomology prove Cartan's Semisimplicity Criterion?
My favorite proof is showing that the trivial module is projective in the category $\mathcal R$ of finite dimensional reps; you can do this using cohomology and the quadratic Casimir (so here the Killing form comes in). After that, showing that all fin. dim. reps. are semisimple amounts to showing that they are all projective in $\mathcal R$, and this follows from the identity $\hom\_\mathfrak g(M,N)=\hom\_\mathfrak g(k,\hom(M,N))$ and a little cohomological yoga.


