Lie algebra theory without algebra

This is an entirely expository piece: the main results discussed are very wellknown and the approach we take is not really new, although the presentation may be somewhat different to what is in the literature. The author’s main motivation for writing this piece comes from a feeling that the ideas deserve to be more widely known. Let g be a Lie algebra over R or C. . A vector subspace I ⊂ g is an ideal if [I,g] ⊂ I. The Lie algebra is called simple if it is not abelian and contains no proper ideals. A famous result of Cartan asserts that any simple complex Lie algebra has a compact real form (that is to say, the complex Lie algebra is the complexification of the Lie algebra of a compact group). This result underpins the theory of real Lie algebras, their maximal compact subgroups and the classification of symmetric spaces. In the standard approach, Cartan’s result emerges after a good deal of theory: the Theorems of Engel and Lie, Cartan’s criterion involving the nondegeneracy of the Killing form, root systems etc. On the other hand if one assumes this result known–by some means–then one can immediately read off much of the standard structure theory of complex Lie groups and their representations. Everything is reduced to the compact case (Weyl’s “unitarian trick”), and one can proceed directly to develop the detailed theory of root systems etc. In [2], Cartan wrote J’ai trouvé effectivement une telle forme pour chacun des types de groupes simples. M. H. Weyl a démontré ensuite l’existence de cette forme par une raisonnement général s’appliquant à tous les cas à fois. On peut se demander si les calculs qui l’ont conduit à ce résultat ne pourraient pas encore se simplifier, ou plutôt si l’on ne pourrait pas, par une raissonnement a priori,