Lie algebras contained in Universal Enveloping Algebra

[Couchyam]

I've got a general question about Lie algebras, which is basically this:
Q: What is there to be said about the Lie algebras that can be identified in the universal enveloping algebra of a particular Lie algebra?
E.g. if I have a Lie algebra of the form
[A,B] = ηB,
then I would like to identify all of the Lie algebras that can be found in the algebra generated by A and B with the relation
AB - BA = ηB.

I would be very happy if someone could direct me to any resources related to this question. I am sorry that the question is somewhat vague, but if something resembling an answer forms in the back of your mind please don't hesitate to post.

 

[fresh_42]

If I understand this correctly, then you look for all isomorphic copies of $\mathfrak{g}$ in $\mathfrak{U(g)}$. I'm not sure, but maybe PBW or one of its corollaries could help.