I neeed to know if the $L^2$ operator
$$ L^2= -\frac{\hbar^2}{\sin \nu} \left(\frac{\partial}{\partial \nu} \left(\sin \nu \frac{\partial}{\partial\nu}\right) + \frac{1}{\sin \nu} \frac{\partial^2}{\partial \phi^2}\right) $$
commutes with the following Hamiltonian, the harmonic oscillator one
$$ H=-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial y^2}+\frac{1}{2}mw^2y^2\psi $$
where $\psi=\psi(y,\nu,\phi)$, that is, y is the radial coordinate and we are in spherical coordinates. I also need to know if $L^2$ also commutes with $y^3$.
My problem here is working with the spherical coordinate operators. I'm looking for the commutation relations for spherical coordinates in the literature, but so far have found nothing. I'm trying to convert this into cartesian coordinates x,y,z, but the task is not that simple. I think that they will commute because the $L^2$ operators only acts in the angle space, but I needed to know the proof for research purposes. Any help is welcome.