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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.

As another explicit example, if you take $G$ acting on $X = G$ by left-translation, then the Lie algebra of symmetries is the right-invariant vector fields on $G$, and there is a unique (flat) connection $\nabla$ on $TG$ whose parallel sections are the right-invariant vector fields. In this case, $V_i = TG\otimes \Lambda^i(T^*G)$ and $D_i$ is just the $\nabla$-twisted exterior derivative $\mathrm{d}^\nabla:TG\otimes \Lambda^i(T^*G)\to TG\otimes \Lambda^{i+1}(T^*G)$, so $r_i = n{n\choose i}$ and the order of $D_i$ is $1$ for all $i$.

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.)

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.

As another explicit example, if you take $G$ acting on $X = G$ by left-translation, then the Lie algebra of symmetries is the right-invariant vector fields on $G$, and there is a unique (flat) connection $\nabla$ on $TG$ whose parallel sections are the right-invariant vector fields. In this case, $V_i = TG\otimes \Lambda^i(T^*G)$ and $D_i$ is just the $\nabla$-twisted exterior derivative $\mathrm{d}^\nabla:TG\otimes \Lambda^i(T^*G)\to TG\otimes \Lambda^{i+1}(T^*G)$, so $r_i = n{n\choose i}$ and the order of $D_i$ is $1$ for all $i$.

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.) One general purpose reference that treats this topic from a much more general point of view (and contains the important references to the earlier literature from the 50s, 60s, and 70s) is Chapter X of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths (Mathematical Sciences Research Institute Publications, volume 18, 1991). It has long been out of print, but you can download a copy of the whole book from http://library.msri.org/books/Book18/MSRI-v18-Bryant-Chern-et-al.pdf. [Chapters IX and X can be read independently from the first eight chapters.] (Caution: There are many misprints, but there was never a second edition published, so the reader has to be somewhat careful.)

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.)

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.

As another explicit example, if you take $G$ acting on $X = G$ by left-translation, then the Lie algebra of symmetries is the right-invariant vector fields on $G$, and there is a unique (flat) connection $\nabla$ on $TG$ whose parallel sections are the right-invariant vector fields. In this case, $V_i = TG\otimes \Lambda^i(T^*G)$ and $D_i$ is just the $\nabla$-twisted exterior derivative $\mathrm{d}^\nabla:TG\otimes \Lambda^i(T^*G)\to TG\otimes \Lambda^{i+1}(T^*G)$, so $r_i = n{n\choose i}$ and the order of $D_i$ is $1$ for all $i$.

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.)

To add to what Ben said: Typically the resolution of the sheaf mapping ${\frak{g}}\to\Gamma(TX)$ is the Spencer resolution, i.e., a canonical $G$-equivariant exact sequence of differential operators $D_i:\Gamma(V_i)\to \Gamma(V_{i+1})$ for $0\le i\le n$ 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 $3$-dimensional Riemannian manifold of constant curvature, one finds that the ranks $r_i$ of the bundles $V_i$ are $$ (r_0,r_1,r_2,r_3) = (3,6,6,3) $$ and the order of the differential operators $D_0$ and $D_2$ is $1$ while the order of the differential operator $D_1$ is 2. If $G$ is the group of conformal transformations of flat $3$-space, 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.)

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.)