The angular momentum operator $J$ in quantum mechanics with the commutation relation
\begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*}
has the structure of a Lie-algebra. It is easy to see that this commutation relation holds in cartesian coordinates. However, using, e.g., the cylindrical coordinate basis $\{{\bf e}_\rho\,{\bf e}_\varphi,{\bf e}_z\}$, the angular momentum has the form
\begin{equation*} {\bf J}=i\hbar\left(\frac{z}{\rho}\partial_\varphi{\bf e}_\rho + (\rho\partial_z-z\partial_\rho){\bf e}_\varphi+\partial_\varphi{\bf e}_z\right). \end{equation*}
Inserting the components naively in this basis in the mentioned commutation relation, one can see that it does not hold. How to properly transform the commutation relation in curvilinear coordinates so that the commutation relation holds? (Changing die Levi-Civita symbol by a factor $\sqrt{g}$ is obviously not enough.)