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 over a field of characteristic 0:

1. is semisimple.

2. has no nonzero Abelian ideal.

3. has zero ideal radical (the radical is the biggest solvable ideal).

4. Every representation of is fully reducible, i.e., is a sum of irreducible representations.

5. 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 ).

Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations. New York: Springer-Verlag, 1984.

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Weisstein, Eric W. "Semisimple Lie Algebra." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SemisimpleLieAlgebra.html