A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a finite-dimensional algebra L over a field of characteristic 0: 1.L is semisimple. 2.L has no nonzero Abelian ideal. 3.L has zero ideal radical (the radical is the biggest solvable ideal). 4. Every representation of L is fully reducible, i.e., is a sum of irreducible representations. 5.L is a (finite) direct product of simple Lie algebras (a Lie algebra is called simple if it is not Abelian and has no nonzero ideal !=L).

semisimple Lie group | simple Lie algebra