Inspired by @ChoF and @annie heart's exploration, I like to post a similar but possibly richer structure related to the decompositions of $U(3)$ or $SU(3)$ representation multiplications $$3 \otimes \bar 3 = 1 \oplus 8.$$ 3 is in the fundamental and 8 is in the adjoint representation of $SU(3)$. Below I modify and preserve the structure of questions from (Subgroups and invariants in a unitary group U(3))
Let $$G=U(3),$$ be the unitary group. Here we consider $G$ in terms of the fundamental representation of U(3). Namely, all of $g \in G$ can be written as a rank-3 (3 by 3) matrices. Now we take a set of $P$ matrices from Gell-Mann matrices.
Can we find some subgroup of Lie group, $$k \in K \subset G= U(3) $$ such that
$$ k^\dagger \{\pm P_1, \pm P_2,\pm P_3,\pm P_4,\pm P_5, \pm P_6, \pm P_7, \pm P_8 \} k$$ $$=\{\pm P_1, \pm P_2,\pm P_3,\pm P_4,\pm P_5, \pm P_6, \pm P_7, \pm P_8\}. $$ This means that the full set $\{\pm P_1, \pm P_2,\pm P_3,\pm P_4,\pm P_5, \pm P_6, \pm P_7, \pm P_8\}$ is invariant under the transformation by $k$. Here $$k^\dagger \equiv (k^*)^T$$ is the complex conjugate ($*$) transpose ($T$) of $k$. What is the full subset (or subgroup) of $K \subset G$?
Here we define: $$ P_1 = \left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right),\;\;\;\; P_2 = \left( \begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right),\;\;\;\; P_3 = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) $$ $$ P_4 = \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ \end{array} \right),\;\;\;\; P_5 = \left( \begin{array}{ccc} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \\ \end{array} \right),\;\;\;\;$$ $$P_6 = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array} \right),\;\;\;\; P_7 = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \\ \end{array} \right).$$ $$P_8 =\frac{1}{\sqrt 3} \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \\ \end{array} \right).$$
This means that $$k^\dagger P_a k= \pm P_b$$ which may transform $a$ to a different value $b$, where $a,b \in \{1,2,3,4,5,6,7,8 \}$. But overall the full set $ \{\pm P_1, \pm P_2,\pm P_3,\pm P_4,\pm P_5, \pm P_6, \pm P_7, \pm P_8 \}$ is invariant under the transformation by $k$.
Please determine the complete $K$.
p.s. I am not so sure the $P_8$ is in the most symmetric form in accordance with $P_j$ $j \in \{1,2,3,4,5,6,7 \}$. So maybe one can suggest the possible modification of $P_8$ to have a larger $K \subset U(3)$