A detailed consideration of the First and Second Universal Enveloping Algebras of a semi-simple Lie algebra and their contributions to infinite-dimensional representations of the group are recently undertaken. Hopefully, the second Universal Enveloping Algebra of a semi-simple Lie algebra would make the classification of these representations complete.
I suggest to read https://www.sciencedirect.com/topics/mathematics/universal-enveloping-algebra
http://www.pitt.edu/~bion/pdf/mehrle-thesis.pdf
https://www.ams.org/journals/tran/1951-070-01/S0002-9947-1951-0044515-0/S0002-9947-1951-0044515-0.pdf
I suggest to read https://www.researchgate.net/deref/https%3A%2F%2Fwww.sciencedirect.com%2Ftopics%2Fmathematics%2Funiversal-enveloping-algebra
Infinitesimal theory of representations of semisimple Lie groups by V. S. Varadarajan
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