I have the following question regarding proof of a theorem in the book "Shlomo Sternberg, Lectures on differential geometry". On page 339 is the following theorem given: Theorem 3.3: Let G be a group of finite type. A necessary and sufficient condition for a G-structure to be locally flat is that the structure-function of all prolongations of G be constant and equal to the corresponding structure constants of the flat G-structure.
The proof for sufficiency is the following: At the k-th stage we get two complete parallelisms with the same constant structure function. We now simply apply Theorem 2.4 of Chapter V. Here the group is the group of all automorphisms of the flat G-structure and the manifold is the kth prolongation of $B_g$.
Theorem 2.4 of Chapter V : Let $G$ be a Lie group with left-invariant forms $\omega^{1}, \ldots, \omega^{n}$, and constants of structure $c_{j k}^{i}$. Let $M$ be a differentiable manifold and $\theta_{1}, \ldots, \theta_{n}$ forms on $M$ satisfying $$ d \theta^{i}=\sum c_{j k}^{i} \theta^{j} \wedge \theta^{k} $$ Then every $y \in M$ has a neighborhood $U$ and a diffeomorphism $f$ of $U \rightarrow G$ so that $$ \omega^{i}=f^{*}\left(\theta^{i}\right) $$ Any two such maps differ by left translation. If $M$ and $G$ are simply connected then $M$ is diffeomorphic to $G$ .
My question is what are the forms that I need to identify and why do we just consider the k-th stage?