Proposition \[P:centralizer2\] is bi-sided. Besides showing many groups are unacceptable using it, we also use it to show $\SO(4)\cong\Sp(1)^{2}/\langle(-1,-1)\rangle$ is acceptable based on the acceptability of $\G_2$.

A critical part in the process of showing Theorem \[T1\] is the study of acceptability/unacceptability of rank $3$ compact simple Lie groups. Among these it is well known that $\SU(4)$, $\SO(7)$, $\Sp(3)$ are acceptable, and Larsen showed that $\PSU(4)$ is unacceptable in [@Larsen1]. The remaining ones are $\SU(4)/\langle-I\rangle\cong\SO(6)$, $\Spin(7)$, $\PSp(3)$. The unacceptability of $\SO(6)$ is first shown by Matthew Weidner in [@Weidner] (cf. Example \[E:SU4\]). The unacceptability of $\Spin(7)$ is due to Gaetan Chenevier and Wee Teck Gan (cf. Theorem \[T:unacceptable-ChenevierGan\]). We show that $\PSp(3)$ is unacceptable (cf. Example \[E:Sp3\]). Curiously, the unacceptability of all of the groups $\SU(4)/\langle-I\rangle\cong\SO(6)$, $\Spin(7)$, $\PSp(3)$ are shown by constructing element-conjugate but not globally conjugate homomorphisms from $(C_4)^{2}$ to them.

Another crucial step in showing Theorem \[T1\] is to show that the group $$\Sp(1)^{3}/\langle
(1,-1,-1), (-1,1,-1)\rangle$$ and groups of the form $$\Sp(1)^{m}/\langle(-1,\dots,-1)\rangle$$ ($m\geq 3$) are unacceptable (cf. Theorem \[T:3A1\] and Example \[E:Sp3-2\]).

[*Notation and conventions.*]{} Let $G$ be a group. For $x,y\in G$, we write $x\sim y$ if $y=
\Ad(g)x=gxg^{-1}$ for some $g\in G$, and we say $x$ 