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.
Read more about this topic: Sl2-triple
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