This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits.
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 simple sl2-module with highest weight j.
The element h of the sl2-triple is semisimple, with the simple eigenvalues j, j − 2, ..., −j on a submodule of g isomorphic to Vj .
Moreover, the direct sum of the eigenspaces of h with non-negative eigenvalues is a parabolic subalgebra of g with the Levi component g0.