Reference request : manifolds and transitive lie group actions I would like to show that any manifold $M$ with a transitive ation from a Lie group $G$ is diffeomorphic to $G/H$ where $H$ is the stabilizer of an elemen...--prophetes.ai

Reference request : manifolds and transitive lie group actions I would like to show that any manifold $M$ with a transitive ation from a Lie group $G$ is diffeomorphic to $G/H$ where $H$ is the stabilizer of an element in $M$. Do you know any reference where I could find at best a detailed proof or at least a sketch of the proof? Thanks.

There is a detailed proof in

Lee, J. M., _Introduction to smooth manifolds_ , Second Edition, Springer (2013).

The result you are looking for is Theorem 21.18 on page 552.