EDIT(3): I am looking for a basis for the Lie algebra of polynomial vector fields on $S^3$.
EDIT(2): I am fairly certain now that my question is more along the lines of, what does the Lie algebra of polynomial vector fields on $S^3$ look like?
EDIT: my question is really the following. The Lie algebra of the diffeomorphism group of $S^1$ is the Witt algebra. What is the corresponding Lie algebra for $S^3$?
(1) What is the diffeomorphism group of the 3-sphere? My reason for asking is that I want to know if there is an analogue of the Witt (=centerless Virasoro) algebra in three dimensions. I am aware of the $W_n$ series in Cartan's classification, but this is not the generalization I am looking for.
(1.0) Alternatively/equivalently, I would like to know how to describe "regular" (smooth?) sections of the tangent bundle on $S^3$ - it seems like it might be helpful (to me, at least) to think of the fibers as copies of $sl(2)(\cong su(2))$. It's been a while since I looked at principal fiber bundles, but this definitely reminds me of one.
(0) Currently I'm trying to think of all of this stuff in terms of [unit] quaternions. This seems promising to me for a number of reasons, so if you can tell me anything about the above in such terms or point me towards papers/preprints/steles about the above in quaternionic language, that would be fantabulous.
(Obligatory "sorry if this is vague to the point of madness" and "sorry if this has been answered previously; I did my best to check the related questions.")