sl2 triple

Summary

An $\mathfrak{sl}_2$-triple in a Lie algebra is a triple of elements $(h, e, f)$ that satisfy the bracket relations of the standard basis of $\mathfrak{sl}_2$. Such a triple spans a subalgebra that is a homomorphic image of $\mathfrak{sl}_2$, so the representation theory of $\mathfrak{sl}_2$ applies to it. In the structure theory of semisimple Lie algebras, each root gives an $\mathfrak{sl}_2$-triple. The Jacobson–Morozov theorem attaches such triples to nilpotent elements.