Starting with Klein geometry I have read that in Klein's approach, geometry can be seen as the study of invariants under a group of transformations. We begin with a space $X$ and a subgroup $G$ of all its bijections (tough usually it would be shrink to a subgroup of the homeomorphisms or diffeomorphisms of $X$). This is called the principal group. Next step is to fix a subspace $S \subseteq X$ and look at
$$
H:=Stab_G (S) \subseteq G,
$$
the subgroup of $G$ that leaves invariant the configuration $S$. Klein's idea was that the space
$$
G/H
$$
is interesting, and this kind of spaces are called homogeneous spaces or Klein geometries. In particular, if the action is transitive and we have $S=\{x\}$ for $x\in X$,
$$
X\approx G/H
$$
that is, we recover the set of points. But  I have been told that we also recover the geometric properties. How that can be done from here?
I'd like to see it with an example. If I consider $X$ as the Euclidean space, an arbitrary point $P\in X$ and the rigid motions $E(3)$ as the principal group $G$, then $H=O(3)$ and it is clear to me that $X=G/H=\mathbb{R}^3$. But how can we deduce from here the gist of this geometry, that is, the Euclidean distance formula or the scalar product? Is there any general method that can be used with any Klein geometry to deduce its "metric" or something?
 A: I have found that the key is in the adjoint representation of $H$ over $\mathfrak{g}$. Because in this context we can look for invariant subspaces and I guess that "inside" is hidden the information for the geometry. I have seen a good explanation in Parabolic Geometries, by Andreas Cap and  Jan Slovak, page 6, but I don't fully understand yet.
I have also found useful the foreword of Differential geometry, Cartan's generalization of Klein Erlangen program, by Sharpe and Chern; although I cannot see yet how the Maurer-Cartan form let us recover the metric.
Edited
After a stop I have returned to the topic, and I have found that the whole chapter 1 of Parabolic Geometries is devoted to this issue, specifically section 1.4. When I finish reading I will edit again to write a summary, or an outline.
On the other hand, I think that the book of Sharpe is the typical book it is worth reading once you know about the topic, because you can think a lot of important stuff, but you need to have a minimum of ideas in your mind to be able to follow it.
