Semisimple Lie Algebra

A semisimple Lie algebra is a finite-dimensional Lie algebra over a field of field characteristic 0 whose Killing form is nondegenerate. For a finite-dimensional Lie algebra over such a field, the following properties are equivalent. Here a Lie ideal is a subspace satisfying .

1. is semisimple.

2. has no nonzero Abelian Lie ideal.

3. has zero radical, where the radical is its largest solvable Lie ideal.

4. Every finite-dimensional Lie algebra representation of is completely reducible, i.e., a direct sum of irreducible Lie algebra representations.

5. is a finite direct product of simple Lie algebras (a Lie algebra is called simple if it is not Abelian and its only Lie ideals are 0 and itself).