TITLE: Polynomial Vector Fields on the 3-Sphere
QUESTION [5 upvotes]: 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.")

REPLY [5 votes]: Complementing André's answer, here's another possible definition of polynomial vector fields on $S^3$.  You think of $S^3$ as the unit sphere in $\mathbb{R}^4$ and consider polynomial vector fields on $\mathbb{R}^4$ which are tangent to the sphere; that is, which annihilate the function $\sum x_i^2$.
One thing to point out, which may or may not be relevant to the applications you have in mind but which I mention since you did mention the Virasoro algebra, is that the structure of the diffeomorphism algebras (or algebras of polynomial vector fields) in dimension greater than 1 is very different than in dimension 1.  For example, you don't have a nice decomposition such as the one
$$
\mathfrak{Vir} = \mathfrak{Vir}^- \oplus \mathfrak{Vir}^0 \oplus \mathfrak{Vir}^+
$$
for the Virasoro algebra, and this in turn hinders the construction of positive energy representations.
I am aware on some work on this topic in the mathematical physics literature; e.g.,
Fock space representations of the algebra of diffeomorphisms of the $N$-torus, F Figueirido and E Ramos
and also papers by TA Larsson.