then $K'=\Ad(g)K$ for some $g\in G^{0}$.

-   for any compact group $\Gamma$ and homomorphism $\phi:\Gamma\rightarrow G$, there exists $g\in G$ such that $\phi(\Gamma)\subset\Ad(g)K$.

-   for any compact group $\Gamma$ and homomorphisms $\phi_1,\phi_2:\Gamma\rightarrow K$, if there exists $g\in G$ such that $\phi_{2}(x)=\Ad(g)(\phi_{1}(x))$ for any $x\in\Gamma$, then there exists $k\in K$ such that $\phi_{2}(x)=\Ad(k)(\phi_{1}(x))$ for any $x\in\Gamma$.

By conditions (2) and (3) in the above, $G$ is acceptable (or strongly acceptable) if and only if $K$ is. Thus, it suffices to study acceptability of compact Lie groups.

[*SCF pairs.*]{} The notion of strongly controlling fusion is invented by Griess ([@Griess]). Considering only homomorphisms from a compact group, we give the following definition.

\[D:SCF\] Let $H$ be a closed subgroup of a real Lie group $G$. We say $H$ strongly controls its fusion in $G$, if for any compact group $\Gamma$ and homomorphisms $\phi_1,\phi_2:\Gamma\rightarrow K$, whenever there exists $g\in G$ such that $\phi_{2}(x)=\Ad(g)(\phi_{1}(x))$ for any $x\in\Gamma$, then there exists $h\in H$ such that $\phi_{2}(x)=\Ad(h)(\phi_{1}(x))$ for any $x\in\Gamma$.

For short, in the below we say $H$ is an “SCF" subgroup of $G$ if $H$ strongly controls its fusion in $G$, and we call $H\subset G$ an “SCF” pair. The condition (3) for a maximal 