This question is a refined version of Representations of infinite dimensional Lie algebras as vector fields on manifolds

I'm interested in the finite dimensional homogeneous spaces of $Diff(S^1)$. There are several papers concerned with *infinite* dimensional homogeneous spaces such as $Diff(S^1)/S^1$, but I can't find anything of the finite dimensional spaces, except the paper by Cartan from 1905, as hinted by Robert Bryant in the above link... From R.B's reply in the link:

*"You could try É. Cartan's papers on infinite pseudogroups (mostly appearing 1904-05). In particular, see Paragraph 57 of Sur la structure des groupes infinis de transformation (suite). There, for example, he proves that the (pseudo-)group of diffeomorphisms of the line has three distinct (i.e., nonequivalent) 2-dimensional homogeneous spaces and seven distinct 3-dimensional homogeneous spaces, etc."*

Basically I'm interested in the two dimensional spaces... it's just that I don't know French, and the exposition in Cartan's paper seems quite brief anyway. **Surely there must be more modern works about the subject?**

Also, it's not very clear to me how to construct these homogeneous spaces as cosets...

EDIT: Here's an example from Cartan's 1905 paper (I still don't know French but I was able to decipher that much):

Suppose we have vector fields as a Lie algebra of $Diff(\mathbb R)$ (Cartan is doing it on the line, but the circle is similar),

$l_n = x^{n+1} \partial_x$.

Cartan obtained corresponding vector fields on three dimensional homogeneous spaces of $Diff(\mathbb R)$. On one of the seven homogeneous spaces they are

$l_n = x^{n+1}\partial _x-(n+1) x^n y\partial _y-(n+1)(n+x z)x^{n-1}\partial _z$,

but what is the homogeneous space (by which I mean, can it be expressed as a quotient)?!

Here's a link to to the relevant pages in Cartan's paper: http://goo.gl/bJXfm

EDIT: Fixed a misunderstanding regarding Cartan's notation in the formula (see RB's answer below). I expanded the vector fields in a basis by Taylor expanding the function $f(x) = \sum c_n x^{n+1}$. That way it's easy to see that they both satisfy the Witt algebra, $[l_n, l_m] = -(n-m)l_{n+m}$...