Jacobson–Morozov theorem
In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after , . Statement The statement of Jacobson–Morozov relies on the following preliminary notions: an sl2-triple in a semi-simple Lie algebra (throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras . Equivalently, it is a triple of elements in satisfying the relations An element is called nilpotent, if the endomorphism (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl2-triple , e must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element can be extended to an sl2-triple. For , the sl2-triples obtained in this way are made explicit in . The theorem can also be stated for linear algebraic groups (again over a field k of characteristic zero): any morphism (of algebraic groups) from the additive group to a reductive group H factors through the embedding Furthermore, any two such factorizations are conjugate by a k-point of H. Generalization A far-reaching generalization of the theorem as formulated above can be stated as follows: the inclusion of