Lie algebras contained in Universal Enveloping Algebra

Couchyam
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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.
 
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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.
 

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