I'm trying to understand the Cartan formalism in the context of General Relativity.
As I understand it given a pseudo-Riemannian spacetime manifold $M$ we can consider the group of spactime transformations $G$ which act transitively on it locally (in this case $SO(3,1)\rtimes\mathbb{R}^{3,1}$). We can also consider the sub-group $H$ (in our case $SO(3,1)$) which is the stabilizer of $M$ such that the quotient of the two is a coset/homogenous space:
$$\frac{G}{H}=\mathbb{R}^{3,1}$$
Now we have a connection 1-form $\omega\in h$, the Lie algebra of $H$ and solder form $\theta$ such that $\left(\theta,\omega\right)\in G$. Our tetrad/vierbein $e$ then acts as a map relating the tangent spaces at a point $p$ $T_{p}()$:
$$e:T(M)\rightarrow T(\frac{G}{H})$$
And we end up with the familiar relation:
$$g_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{b}\eta_{ab}$$
It would seem by this construction that the homogenous “model” space would always have to be $\mathbb{R}^{3,1}$ for a pseudo-Riemannian spacetime; however in perusing the method of moving frames it's stated that:
“When the homogenous space is a quotient of special orthogonal groups, this reduces to the standard conception of a vierbein. “
Which would seem to imply our homogenous space should be something more like a sphere. Can someone please explain what I'm missing here?