What are the conformal killing vector fields on $S^2$ equipped with the round metric? (expressed in the standard coordinates $(\theta, \phi)$) The space of conformal killing vector fields make a 6 dimensional vector space, so I am asking for a simple basis expressed in the standard coordinates. (I know $\sin \phi \partial_{\phi}$ is one of them, which is conformal Killing but not Killing).
The space of conformal diffeomorphisms of $S^2$ is known to be the set of Mobius transformations, which is a 6-parameter family. Is there a way to compute the conformal killing vector fields from the Mobius transformations?
If there is a reference you know that lists them, please share them.