I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 2. It this proposition:
My problem is that I don't understand the converse of this proposition. Is not clear how he is doing that linear change of coordinate, nothing is clear or formal and I'm also new to the subject. Can some one fill in some details for me please? Or maybe I should check another book on $G$- structures. Can some one point me to a good book on this subject?
EDIT: this is the definition of the integrability that is used here.
