Linear map connecting two left-invariant one-forms valued in different Lie algebras

Question

How to see that a left-invariant one-form on a Lie group valued in a different Lie algebra can be factorized through the canonical left-invariant Maurer-Cartan form of this Lie group followed by a linear map between the two Lie algebras? In particular, how to see the existence and possibly uniqueness of this linear map? Thanks.

Answer 1

If you have a form $\phi\in\Omega^1(G,\mathfrak h)$, then $\phi(e)$ by definition is a linear map $T_eG=\mathfrak g\to \mathfrak h$. The only reasonable notion of equivariancy in this general setting is that for each $g\in G$ with left translation $\lambda_g:G\to G$ one has $(\lambda_g)^*\phi=\phi$. But for $\xi\in T_gG$, you then get $\phi(g)(\xi)=((\lambda_{g^{-1}})^*\phi)(g)(\xi)=\phi(e)(T_g\lambda_{g^{-1}}\cdot\xi)$. So this shows that $\phi=\phi(e)\circ\omega$, where $\omega$ is the left Maurer Cartan form on $G$.