In mathematics, a semisimple element is an abstract element of an algebraic structure that generalizes a diagonalizable matrix. A precise meaning depends on context: A semisimple element in the endomorphism ring of a vector space is a semisimple operator. In a semisimple Lie algebra, an element is semisimple if its image under the adjoint representation is semisimple; see Semisimple Lie algebra#Jordan decomposition.