Let $U(\mathfrak{sl}_2)$ be the Universal Enveloping Algebra of $\mathfrak{sl_2}$ over a field $K$, i.e. the (non-commutative) algebra generated by three generators $E,F,H$ subject to the commutator relations
$$[E,F] = H, \quad [H,E] = 2E,\quad [H,F] = -2F.$$
What are the units of $U(\mathfrak{sl}_2)$?
In view of the natural inclusion $\iota \colon K \to U(\mathfrak{sl}_2)$ and the projection $\pi \colon U(\mathfrak{sl}_2) \to K$ (sending $E,F,H$ to $0$), it is clear that all units $u$ of $K$ yield units $\iota(u)$ of $U(\mathfrak{sl}_2)$, and all units $v \in U(\mathfrak{sl}_2)$ map to units in $K$ under $\pi$, hence are of the form $v = \iota(\pi(v)) +p(E,F,H)$, where $p$ is a polynomial expression without constant term (this makes use of the Poicare-Birkhoff-Witt Theorem), and $\iota(\pi(v))$ is the image of a unit in $K$.