is defining a kind of “strongly acceptability" by allowing homomorphisms from any compact group.

\[D:acceptable1\] A real Lie group $G$ with finitely many connected components is called acceptable if for any finite group $\Gamma$ and every pair of element-conjugate homomorphisms $\phi_1,\phi_2:\Gamma\rightarrow G$, $\phi_1$ and $\phi_2$ are globally conjugate.

A real Lie group $G$ with finitely many connected components is called strongly acceptable if for any compact group $\Gamma$ and every pair of element-conjugate homomorphisms $$\phi_1,\phi_2:\Gamma\rightarrow G,$$ $\phi_1$ and $\phi_2$ are globally conjugate.

\[D:acceptable2\] A real Lie group $G$ with finitely many connected components is called unacceptable if there exists a finite group $\Gamma$ and two element-conjugate homomorphisms $\phi_1,\phi_2:\Gamma\rightarrow G$, $\phi_1$ and $\phi_2$ are not globally conjugate.

A real Lie group $G$ with finitely many connected components is called strongly unacceptable if there is a compact group $\Gamma$ and two element-conjugate homomorphisms $\phi_1,\phi_2:\Gamma\rightarrow G$, $\phi_1$ and $\phi_2$ are not conjugate in image.

Maximal compact subgroups and “SCF” subgroups
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[*Maximal compact subgroup.*]{} Any Lie group $G$ with finitely many connected components has a maximal compact subgroup. A maximal compact subgroup $K$ has the following property ([@Borel Page 124, Theorem 1.2 and Corollary 1.3]):

-   if $K'$ is another maximal compact subgroup, 