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 ).

See also Semisimple Lie Group ,

Simple Lie Algebra
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References Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations. New York: Springer-Verlag,
1984. Referenced on Wolfram|Alpha Semisimple Lie Algebra
Cite this as:
Weisstein, Eric W. "Semisimple Lie Algebra."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/SemisimpleLieAlgebra.html

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