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 if $A\in \textrm{O}(3,\mathbb{R})$ and $\det(A)=-1$, there exists a $U\in \textrm{SU}(3)$ such that $$UAU^{\dagger} = \left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & e^{i\theta} & 0 \\ 0 & 0 & e^{-i\theta} \\ \end{array}\right)$$ with some $\theta\in\mathbb{R}$. Then there exists a $V\in \textrm{SU}(2)$ such that $$V\left(\begin{array}{cc} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \\ \end{array}\right)V^{\dagger} = \left(\begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array}\right)$$ So if you define $$W = \left(\begin{array}{cc} 1 & 0 \\ 0 & V \\ \end{array}\right)U$$ then $WAW^{\dagger}$ will be of such form that reflection and rotation are clearly carried out with respect to the same axis. Only problem is that $W$ doesn't necessarily have only real entries. How to prove that $W$ is necessarily proportional to a real matrix?