Let $R_\theta$ denote the appropriate element of $SO(2)$, as is conventional. Let $n$ be a positive integer, and consider $D = \text{diag}(R_{\theta_1}, \dots, R_{\theta_n}) \in SO(2n)$. For simplicity, assume the $\theta_i$ are in $[0, 2\pi)$ and are distinct.
Question: If there exists an orthonormal basis, $\beta$, of $\mathbb{R}^{2n}$ such that \begin{equation*} M_\beta(D) = g^{-1}Dg = \text{diag}(R_{\psi_1}, \dots, R_{\psi_n}), \end{equation*} where the $i^{th}$ column of $g$ is the $i^{th}$ vector of $\beta$, and $\psi_i \in [0, 2\pi)$, then is it true that $\left\{ \theta_i \right\} = \left\{ \psi_i \right\}$?
Many thanks!
Consider $n=2$ and $g=g^{-1}=\pmatrix{0&I_2\\ I_2&0}$.