Editing Semisimple Lie algebra (section)

== Basic properties ==
*Every ideal, quotient and product of semisimple Lie algebras is again semisimple.<ref>{{harvnb|Serre|2000|loc=Ch. II, § 2, Corollary to Theorem 3.}}</ref>
*The center of a semisimple Lie algebra <math>\mathfrak g</math> is trivial (since the center is an abelian ideal). In other words, the [[adjoint representation of a Lie algebra|adjoint representation]] <math>\operatorname{ad}</math>  is injective. Moreover, the image turns out<ref>Since the Killing form ''B'' is non-degenerate, given a derivation ''D'', there is an ''x'' such that <math>\operatorname{tr}(D\operatorname{ad}y) = B(x, y)</math> for all ''y'' and then, by an easy computation, <math>D = \operatorname{ad}(x)</math>.</ref> to be <math>\operatorname{Der}(\mathfrak g)</math> of [[derivation (abstract algebra)|derivations]] on <math>\mathfrak{g}</math>. Hence, <math>\operatorname{ad}: \mathfrak{g} \overset{\sim}\to \operatorname{Der}(\mathfrak g)</math> is an isomorphism.<ref>{{harvnb|Serre|2000|loc=Ch. II, § 4, Theorem 5.}}</ref> (This is a special case of [[Whitehead's lemma (Lie algebras)|Whitehead's lemma]].)
*As the adjoint representation is injective, a semisimple Lie algebra is a [[linear Lie algebra]] under the adjoint representation. This may lead to some ambiguity, as every Lie algebra is already linear with respect to some other vector space ([[Ado's theorem]]), although not necessarily via the adjoint representation.  But in practice, such ambiguity rarely occurs.
*If <math>\mathfrak g</math> is a semisimple Lie algebra, then <math>\mathfrak g = [\mathfrak g, \mathfrak g]</math> (because <math>\mathfrak g/[\mathfrak g, \mathfrak g]</math> is semisimple and abelian).<ref>{{harvnb|Serre|2000|loc=Ch. II, § 3, Corollary to Theorem 4.}}</ref>
*A finite-dimensional Lie algebra <math>\mathfrak g</math> over a field ''k'' of characteristic zero is semisimple if and only if the base extension <math>\mathfrak{g} \otimes_k F</math> is semisimple for each field extension <math>F \supset k</math>.<ref>{{harvnb|Jacobson|1979|loc=Corollary at the end of Ch. III, § 4.}}</ref> Thus, for example, a finite-dimensional real Lie algebra is semisimple if and only if its complexification is semisimple.