To add to what Ben said:  Typically the resolution of the sheaf mapping ${\frak{g}}\to\Gamma(TX)$ is the *Spencer resolution*, which is a canonical $G$-equivariant exact sequence of differential operators $D_i:\Gamma(V_i)\to \Gamma(V_{i+1})$ for $0\le i< n = \dim X$, where $V_0 = TX$ and $\mathrm{ker}(D_0) = {\frak{g}}$.  When you have a flat $(G,X)$-geometry on a manifold $M$, this sequence pulls back (under the local charts) to define the corresponding sequence on $M$.

For example, when $G$ is the group of isometries of a (simply-connected) $3$-dimensional Riemannian manifold $X$ of constant curvature, one finds that the ranks $r_i$ of the bundles $V_i$ over $X$ are 
$$
(r_0,r_1,r_2,r_3) = (3,6,6,3)
$$
and that the order of the differential operators $D_0$ and $D_2$ is $1$ while the order of the differential operator $D_1$ is 2.  When $G$ is the group of conformal transformations of compactified flat $3$-space~$X = S^3 = \mathbb{R}^3\cup\{\infty\}$, then the ranks are
$$
(r_0,r_1,r_2,r_3) = (3,5,5,3)
$$
and the order of the differential operators $D_0$ and $D_2$ is $1$ while the order of the differential operator $D_1$ is 3.

Look in any book that treats the *Spencer resolution* or the topic of *Lie equations* for references.  (I'm traveling, so I don't have access to the literature.)