Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\mathfrak{sl}(2,\mathbb{C})$

In this work we provide the complete classification of all non-zero finite-dimensional complex Lie subalgebras of the universal enveloping algebra of the special linear Lie algebra $\mathfrak{sl}(2,\mathbb{C})$. That is, we show that every such nilpotent Lie subalgebra is abelian and every non-solva