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.

Cartan algebra | Lie algebra | semisimple Lie algebra