Editing Semisimple Lie algebra (section)

== Significance ==
The significance of semisimplicity comes firstly from the [[Levi decomposition]], which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple.

Semisimple Lie algebras have a very elegant classification, in stark contrast to [[solvable Lie algebra]]s. Semisimple Lie algebras over an algebraically closed field of characteristic zero are completely classified by their [[root system]], which are in turn classified by [[Dynkin diagram]]s. Semisimple algebras over non-algebraically closed fields can be understood in terms of those over the algebraic closure, though the classification is somewhat more intricate; see [[real form]] for the case of real semisimple Lie algebras, which were classified by [[Élie Cartan]].

Further, the [[representation theory of semisimple Lie algebras]] is much cleaner than that for general Lie algebras. For example, the [[Jordan–Chevalley decomposition|Jordan decomposition]] in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not the case for Lie algebras in general.

If <math>\mathfrak g</math> is semisimple, then <math>\mathfrak g = [\mathfrak g, \mathfrak g]</math>. In particular, every linear semisimple Lie algebra is a subalgebra of <math>\mathfrak{sl}</math>, the [[special linear Lie algebra]]. The study of the structure of <math>\mathfrak{sl}</math> constitutes an important part of the representation theory for semisimple Lie algebras.