with our method. Our proof of “SCF" property for a pair $H\subset G$ is through studying the intersections $H\cap\Ad(g)H$ ($g\in G$), decomposing the double coset space $H\backslash G/H$, and calculating the centralizer subgroup $Z_{H}(g)=
H\cap Z_{G}(g)$ ($g\in G$). These are of independent interest. Using “SCF" property we show that groups in the items (1)-(4) of Theorem \[T1\] are acceptable. The connection between acceptability and “SCF" property is through the following proposition.

\[P1\] 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 $H$.

In the second aspect, we show many groups are unacceptable. For doing this the following Lemma \[L:product2\] and Proposition \[P:centralizer2\] play important roles, though they are very simple. Lemma \[L:product2\] is clear to see, and Proposition \[P:centralizer2\] is also easy to show.

\[L:product2\] Let $Z\subset Z(G_1)\times Z(G_2)$ with $Z\cap Z(G_1)=Z\cap Z(G_2)=1$. Suppose $(G_1\times G_2)/Z$ is acceptable (or strongly acceptable), then so are $G_1$ and $G_2$.

\[P:centralizer2\] Let $G$ be a compact Lie group and $A\subset G$ be a closed abelian subgroup. If $G$ is strongly acceptable (or acceptable), then so is $Z_{G}(A)$.

Note that the effect of 