A Lie algebra is said to be simple if it is not Abelian and has no nonzero proper ideals. Over an algebraically closed field of field characteristic 0, every simple Lie algebra is constructed from a simple reduced root system by the Chevalley construction, as described by Humphreys. Over an algebraically closed field of field characteristic >7, every simple Lie algebra is constructed from a simple reduced root system (as in the characteristic 0 case) or is a Cartan algebra. There also exist simple Lie algebras over algebraically closed fields of field characteristic 2, 3, and 5 that are not constructed from a simple reduced root system and are not Cartan algebras.

Cartan algebra | Lie algebra | semisimple Lie algebra