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I think that the elements $g = \dfrac1{\sqrt2}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}^{\oplus3}$ and $h = \dfrac1 2\begin{pmatrix} 1 & 1 & 1 & 0 & 1 & 0 \\ -1 & 1 & -1 & 0 & 1 & 0 \\ -1 & 1 & 1 & 0 & -1 & 0 \\ 1 & 1 & -1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{pmatrix}$ in $\operatorname{SO}(6)$ satisfy $g^{-1}\gamma_i g = h^{-1}\gamma_i'h$ for all $i \in \{1, 2, 3\}$.