Sl2-triple
In the theory of Lie algebras, an sl2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra sl2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits. Definition Elements {e,h,f} of a Lie algebra g form an sl2-triple if These commutation relations are satisfied by the generators of the Lie algebra sl2 of 2 by 2 matrices with zero trace. It follows that sl2-triples in g are in a bijective correspondence with the Lie algebra homomorphisms from sl2 into g. The alternative notation for the elements of an sl2-triple is {H, X, Y}, with H corresponding to h, X corresponding to e, and Y corresponding to f. H is called a neutral, X is called a nilpositive, and Y is called a nilnegative. Properties Assume that g is a finite dimensional Lie algebra over a field of characteristic zero. From the representation theory of the Lie algebra sl2, one concludes that the Lie algebra g decomposes into a direct sum of finite-dimensional subspaces, each of which is isomorphic to Vj, the (j + 1)-dimensional