ideal, quotient and product of semisimple Lie algebras is again semisimple. The center of a semisimple Lie algebra is trivial (since the center is an abelian ideal). In other words, the adjoint representation is injective. Moreover, the image turns out to be of derivations on . Hence, is an isomorphism. (This is a special case of 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 is a semisimple Lie algebra, then (because is semisimple and abelian). A finite-dimensional Lie algebra over a field k of characteristic zero is semisimple if and only if the base extension is semisimple for each field extension . Thus, for example, a finite-dimensional real Lie algebra is semisimple if and only if its complexification is semisimple. Jordan decomposition Each endomorphism x of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a semisimple (i.e., diagonalizable over the algebraic closure) and nilpotent part such that s and n commute with each other. Moreover, each of s and n is a polynomial in x. This is the Jordan decomposition of x. The above applies to the adjoint representation of a semisimple Lie algebra . An element x of is said to be semisimple (resp. nilpotent) if is a semisimple (resp. nilpotent) operator. If , then the abstract Jordan decomposition states that x can be written uniquely as: where is semisimple, is nilpotent and . Moreover, if commutes with x, then it commutes with both as well. The abstract Jordan decomposition factors through any representation of in the sense that given any representation ρ, is the Jordan decomposition of ρ(x) in the endomorphism algebra of the representation space. (This is proved as a consequence of Weyl's complete reducibility theorem; see Weyl's theorem on complete reducibility#Application: preservation of Jordan decomposition.) Structure Let be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of can be described by an adjoint action of a certain distinguished subalgebra on it, a Cartan subalgebra. By definition, a Cartan subalgebra (also called a maximal toral subalgebra) of is a maximal subalgebra such that,