Let $G$ be a Lie group. A vector field $X\in\mathfrak{X}(G)$ is left-invariant if the diagram below is commutative:
for every $g\in G$ where $L_g$ stands for the left translation by $g$. Now a differential $1$-form $\omega\in \Omega^1(G)$ is left invariant if $$L^*_g(\omega)=\omega,$$ for every $g\in G$ where $L_g^*(\omega)$ is the pullback form. Is there a way to define a left-invariant form using a commutative diagram as above (maybe using $T^*G$)?