Property (\[Eq1\]), one can show that $\psi'(h)$ does not depend on the choice of the sequence $\{\gamma_{n}\}_{n\geq 1}$. One can show that the map $\psi'$ is an injective homomorphism and it satisfies all required properties.

As $H$ is a compact Lie group and $\psi,\psi'$ are element-conjugate, by the assumption there exists $g\in G$ such that $\psi'(h)=\Ad(g)(\psi(h))$ for any $h\in H$. Then, $\phi_{2}(x)=\Ad(g)(\phi_{1}(x))$ for any $x\in\Gamma_1$. Thus, $\phi_1$ and $\phi_2$ are globally conjugate. Therefore, $G$ is strongly acceptable.

By Lemma \[L:acceptable1\], to check whether a Lie group $G$ with finitely many connected components is strongly acceptable, it suffices to consider element-conjugate homomorphisms from compact Lie groups to $G$.

It looks to us that strong acceptability should be a consequence of acceptability.

\[Q:acceptable1\] Is a Lie group $G$ acceptable if and only if it is strongly acceptable?

We call a compact Lie group $H$ a quasi-torus if $H^{0}$ is a torus. In the Proposition \[P:acceptable2\] below, we show that acceptability implies that element-conjugate homomorphisms from a a quasi-torus are globally conjugate.

\[L:centralizer1\] Let $G$ be a compact Lie group. Then there exist only finitely many conjugacy classes of subgroups $H$ of $G$ of the form $H=Z_{G}(A)$ with $A$ an abelian subgroup of 