and $y$ are conjugate (or $G$-conjugate). For a subgroup $G_1\subset G$ (or an over-group $G_{2}\supset G$), we write $x\sim_{G_1}y$ if $y=\Ad(g)x=
gxg^{-1}$ for some $g\in G_1$ (or $g\in G_2$).

Let $G$ be a connected compact Lie group. Write $Z(G)$ for the center of $G$, and $G_{der}=[G,G]$ for the derived subgroup of $G$.

In this paper we consider only real Lie group with finitely many connected components. This includes particularly all real algebraic groups.

[*Acknowledgement.*]{} I would like to thank Xinwen Zhu for bringing up this question to my attention, and to thank Gaetan Chenevier, Wee Teck Gan, Matthew Weidner for helpful communications. Part of this paper was written when the author visited Paris Diderot University (Paris 7) in May 2018. I would like to thank Huayi Chen for the invitation.

Acceptability conditions
========================

Let $G$ be a real Lie group, and $\Gamma$ be a group. Two homomorphisms $\phi_{1},\phi_{2}:\Gamma
\rightarrow G$ are called element-conjugate if $\phi_{1}(x)\sim\phi_{2}(x)$ for any $x\in\Gamma$. They are called globally conjugate if there exists $g\in G$ such that $\phi_{2}(x)=\Ad(g)(\phi_{1}(x))$ for all $x\in
\Gamma$. They are called conjugate in image if $\phi(2)(\Gamma)=\Ad(g)(\phi_{1}(\Gamma))$ for some $g\in G$.

Following [@Larsen1], we define acceptability and unacceptability as follows. The only new addition 