I need prove that any element of $U(\mathfrak{sl}_2)$ can be represented by linear combination of elements $e^i h^j C^k$, where $C=ef+fe+\dfrac{h^2}{2}$.
$e=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \ \ h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \ \ f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$.
First I think about $U(\mathfrak{sl}_2)$. Using PBW-theorem I know that the elements $e^ih^jf^k$ form a basis of $U(\mathfrak{sl}_2)$. But then $U(\mathfrak{sl}_2)$ is finite dimensional? I have $e^2=0,f^2=0, h^2=1$.