Original setup

Consider the manifold $M=\mathbb R^3$ with the natural vector bundle connection $\nabla$. This connection, like any connection on a vector bundle, induces, or is induced by, a principal connection $\Theta$ on the frame bundle $FM=\mathbb R^3 \times GL(3)$. The curvature of this connection is, of course, zero: $$ \Omega=d\Theta-[\Theta,\Theta]=0. $$

Cartan geometry

I want to obtain the pair $(\mathbb R^3,\Theta)$ from the point of view of Cartan geometry, to understand to what extent the flatness of $\mathbb{R}^3$ can be generalized to arbitrary homogeneous spaces. We have the affine group $G=Aff(3)=\mathbb{R}^3 \rtimes GL(3)$ (we fix it as the "relevant" transformations on $\mathbb R^3$) together with the closed subgroup $H=GL(3)$ (the "relevant" transformations fixing a point). The Klein geometry $(G,H)$ provides the base space $M$, and the Maurer-Cartan form $A$ of $G$ is a Cartan connection with, of course, zero curvature $$ dA-[A,A]=0, $$ because of the Maurer-Cartan equation for any Lie group. Since this is a reductive Klein geometry, the Cartan connection $A$ splits $$ A=A_{\mathfrak{h}}+A_{\mathfrak{p}} $$ for every specific choice of the subspace $\mathfrak{p}\subset \mathfrak{g}$ such that $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{p}$. The part $A_{\mathfrak{h}}$ is a principal connection on the frame bundle $FM$, playing the same role as $\Theta$, but not necessarily equal to $\Theta$ (is a different connection). Moreover, even if $A$ has zero curvature I think we cannot deduce that $A_{\mathfrak{h}}$ has zero curvature.

My questions are:

1. What data am I missing from the original setup to be able to fix a $\mathfrak{p}$ such that $A_{\mathfrak{h}}=\Theta$?
2. In general, what additional requirement do we have to impose to a reductive Klein geometry $(G,H)$ in such a way that we have a canonical choice of $\mathfrak{p}$ giving rise to a principal connection 1-form $\Theta:=A_{\mathfrak{h}}$ which is flat?

The motivation for my questions is that I have read that the Klein geometries are the flat models for the Cartan geometries. But even if I understand that they are flat, in the sense that their natural Cartan connections has zero curvature, I want to know which among them are "really flat", in the sense that they have a natural principal connection with zero curvature.