The space X = G/H of a Klein geometry is a smooth manifold of dimension There is a natural smooth left action of G on X given by Clearly, this action is transitive (take a = 1), so that one may then regard X as a homogeneous space for the action of G. The stabilizer of the identity coset H ∈ X is precisely the group H. Given any connected smooth manifold X and a smooth transitive action by a Lie group G on X, we can construct an associated Klein geometry (G, H) by fixing a basepoint x0 in X and letting H be the stabilizer subgroup of x0 in G. The group H is necessarily a closed subgroup of G and X is naturally diffeomorphic to G/H.
The Klein geometry associated to a homogeneous space X is then unique up to isomorphism (i.e. it is independent of the chosen basepoint x0).
Given a Lie group G and closed subgroup H, there is natural right action of H on G given by right multiplication.
The orbits are simply the left cosets of H in G. One concludes that G has the structure of a smooth principal H-bundle over the left coset space G/H: The action of G on X = G/H need not be effective.
This is the geometry (G0, G0 ∩ H) where G0 is the identity component of G. Note that G = G0 H. A Klein geometry (G, H) is said to be reductive and G/H a reductive homogeneous space if the Lie algebra