projections of $g_{2\theta}$ and $x$ in $G/\langle-1\rangle\cong\SO(8)$ commute. When $\frac{4\theta}{\pi}\not\in
\mathbb{Z}$, $\SO(7)^{g_{2\theta}}=\SO(6)$ is connected. Thus, $$H\cap\Ad(g)H=\Spin(6).$$ Hence, $$g=[g_{\theta}]\in Z_{G'}(H\cap\Ad(g)H)\subset Z_{G'}(H\cap\Ad(g)H)H.$$ When $\frac{4\theta}{\pi}
\in\mathbb{Z}$ (and $\theta\in[0,2\pi)-\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\}$), we have $\theta=
\frac{k\pi}{4}$ with $k=1,3,5,7$. In this case $[g_{2\theta},x]=\pm{1}$ if and only if $x\in\Pin(6)
\subset\Spin(7)$. Thus, $$H\cap\Ad(g)H=\Pin(6).$$ One calculates that $Z_{G'}(\Pin(6))=Z_{G'}$. It is clear that $[g_{\frac{k\pi}{4}}]\not\in Z_{G'}H$ ($k=1,3,5,7$). Thus, $$g\not\in Z_{G'}(H\cap
\Ad(g)H)H$$ for $g=[g_{\frac{k\pi}{4}}]$ ($k=1,3,5,7$).

Remark: after posing a previous version of this paper on the arXiv, Gaetan Chenevier and Wee Teck Gan showed me that ¡°$\Spin(7)$ is unacceptable¡±. This led me to check statements and proofs in that version. Proposition \[P:O(2n+1)-1\] and Proposition \[P:spin7-2\] are the corrected statements.

Acceptable/unacceptable compact Lie groups
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Examples of acceptable compact Lie groups
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The acceptability of groups in items (1) and (2) of the following theorem are well known (cf. [@Larsen1],[@Larsen2],[@Griess]). Here we show their strong acceptability from the stronger “SCF” property. Acceptability of the group $\SO(4)$ is new.

\[T:acceptable1\] Groups isomorphic to any one in the following list are strongly acceptable,

-   $\U(n)$, $\SU(n)$, $\Sp(n)$, $\O(n)$, $\SO(2n+1)$.

-   $\G_2$;

-   $\SO(4)$.

For groups in item (1), strongly acceptability of $\U(n)$ and $\SU(n)$ follow from the character theory of representations of compact Lie groups. By Theorem 