compact subgroup just means a maximal compact subgroup $K$ is an “SCF" subgroup of $G$. Generally we have the following statement.

\[P:SCF\] Let $H$ be a closed subgroup of a real Lie group $G$. Suppose $H$ strongly controls its fusion in $G$. If $G$ is acceptable (or strongly acceptable), then so is $H$.

We show for strongly acceptability. The proof for acceptability is similar. Suppose $G$ is strongly acceptable. Let $\Gamma$ be a compact group and $\phi_1,\phi_2:\Gamma\rightarrow H$ be two element-conjugate homomorphisms. Considering $\phi_1,\phi_2$ as homomorphism to $G$, by the strongly acceptability of $G$ there exists $g\in G$ such that $\phi_{2}(x)=\Ad(g)(\phi_{1}(x))$ for any $x\in\Gamma$. As $H$ is an “SCF" subgroup of $G$, there exists $h\in H$ such that $\phi_{2}(x)=\Ad(h)(\phi_{1}(x))$ for any $x\in\Gamma$. This means that $\phi_1$ and $\phi_2$ are globally conjugate. Thus, $H$ is strongly acceptable.

First reduction
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\[L:acceptable1\] For a compact Lie group $G$ to be strongly acceptable, if and only if for any compact Lie group $H$ and every pair of element-conjugate homomorphisms $\phi_1,\phi_2:H\rightarrow G$, $\phi_1$ and $\phi_2$ are globally conjugate.

The “only if" part is trivial. For the “if" part, suppose the statement above for compact Lie groups $H$ holds. Write $G^{\#}$ for the set of 