= Solution
For $\xi$ in the <Lie algebra> $\mathfrak g$, set $\Phi^t=R_{\exp(t\xi)}$. The <one-parameter subgroup> law gives the flow law, and
$$
\left.\frac d{dt}\right|_{0}g\exp(t\xi)=(dL_g)_e\xi=l_\xi(g),
$$
so this is the global flow of the <left-invariant vector field> $l_\xi$.
For $f(g)=R_g^*\alpha$, every tangent vector at $g$ is the initial velocity of $g\exp(t\xi)$ for some $\xi$. Since $R_{g\exp(t\xi)}=R_{\exp(t\xi)}\circ R_g$,
$$
\left.\frac d{dt}\right|_0f(g\exp(t\xi))
=R_g^*(\mathcal L_{l_\xi}\alpha).
$$
Thus all $\mathcal L_{l_\xi}\alpha$ vanish exactly when every derivative of $f$ vanishes. This is equivalent to $f$ being a <locally constant function>.
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