If you love semisimple Lie algebras, like I do, then you should take a look at Meinolf Geck’s paper

*On the construction of semisimple Lie algebras and Chevalley groups*that just appeared in Proceedings of the AMS (journal version, preprint version). He gives an elementary and canonical construction of a semisimple complex Lie algebra from its root system. It is a pleasing read.He explains that it is a retelling of results on quantum groups from various works by George Lusztig, made simpler by translating them into the context of semisimple Lie algebras.

The construction goes by defining a vector space

*M*with basis corresponding to a Chevalley basis of the Lie algebra — in this language a copy of the root system and a copy of the set of simple roots — and specific elements of*gl(M),*namely one element for each of the root subalgebras corresponding to plus or minus a simple root. He shows that the Lie subalgebra of*gl(M)*generated by these elements is semisimple with root system the one you started with.