## lie algebras – Reference: Compact algae subalgebras of lees

Yes $$mathfrak {g}$$ is a simple and compact Lie algebra, we know a lot about its subalgebras:

• They are all reducers.
• Any simple subalgebra is also compact. These correspond to 1 to 1 with the simple subalgebras of $$mathfrak {g} ^ mathbb {C}$$.
• Any abelian subalgebra is contained in a Cartan subalgebra (that is, it is an abelian algebra containing only semi-simple elements).

What I can not find, however, is an explicit description of what such subalgebras look like. Simple sub-algebras can be quite unpleasantly integrated (for example, in the case of S-subalgebras), and this affects what abelian subalgebras can commute with such a subalgebra.

Has anyone encountered such a description?

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