Editing Semisimple Lie algebra (section)

{{Short description|Direct sum of simple Lie algebras}}
{{Lie groups}}

In [[mathematics]], a [[Lie algebra]] is '''semisimple''' if it is a [[direct sum of modules|direct sum]] of [[Simple Lie algebra|simple Lie algebras]]. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper [[Lie algebra#Subalgebras.2C ideals and homomorphisms|ideals]].)

Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of [[Characteristic (algebra)|characteristic]] 0. For such a Lie algebra <math>\mathfrak g</math>, if nonzero, the following conditions are equivalent:
*<math>\mathfrak g</math> is semisimple;
*the [[Killing form]] <math>\kappa(x, y) = \operatorname{tr}(\operatorname{ad}(x)\operatorname{ad}(y))</math> is [[non-degenerate]];
*<math>\mathfrak g</math> has no non-zero abelian ideals;
*<math>\mathfrak g</math> has no non-zero [[solvable Lie algebra|solvable]] ideals;
* the [[Radical of a Lie algebra|radical]] (maximal solvable ideal) of <math>\mathfrak g</math> is zero.