Proposition 53

Let G be a Lie group, A a complete normable algebra and $\phi$ a Lie group morphism of G into $A^*$. For all $g \in G$, let $\omega_g = \phi(g)^{-1} d_g \phi$. Then $\omega$ is the left invariant differential form on G whose value at e is $d_e \phi$.

We apply Lemma 2 with $E = A$ and $U = G$. The differential at g of the mapping $h \mapsto \phi(g^{-1}h) = \phi(g)^{-1} \phi(h)$ is $\phi(g)^{-1} d_g \phi$.