**Orthogonal complements of semisimple ideals **

There is a general fact about semisimple ideals in arbitrary Lie algebras that we prove next; it will be an application of the material on complete reducibility. We will use it to complete the picture of the abstract Jordan decomposition in a semisimple Lie algebra.

Proposition 1Let be a semisimple ideal in the Lie algebra . Then there is a unique ideal with .

The idea is that we can find a complementary -module with

as -modules. Now because is an ideal and because is stable under the action of . The converse is true: is the centralizer of , and thus an ideal, because anything commuting with can have no part in in the decomposition ( being semisimple). Thus, I hereby anoint with the fraktur font and call it to recognize its Lie algebraness.

Given a splitting as above, would have to be the centralizer of , so uniqueness is evident. (more…)