Consider the 2-torus $T \subset SO(7)$ defined by $T = \left\{ \mathrm{diag}(R_{\theta_1}, R_{\theta_2}, R_{-(\theta_1 + \theta_2)}, 1) \mid \theta_1, \theta_2 \in \mathbb{R} \right\}$, where \begin{equation*} R_\theta = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \end{equation*} is a typical element of $SO(2)$.
Let $A \in T$ such that $\theta \neq 0$ or $\pi$, $\theta_1 = \theta$, and $\theta_2 = -\theta$, so that $-(\theta_1 + \theta_2) = 0$. Let $H_1 = \mathrm{span}\left\{ e_1, e_2 \right\}$ and $H_2 = \mathrm{span}\left\{ e_3, e_4 \right\}$. Then $H_1$ and $H_2$ are stabilized by $A$.
Question: Thinking of $\mathbb{R}^7 = \mathbb{C}^2 \times \mathbb{C} \times \mathbb{R}$, is it true that $H_1$ and $H_2$ are the unique pair of orthogonal (with respect to the inner product in $\mathbb{R}^7$) one-dimensional $\mathbb{C}$-subspaces in $\mathbb{C}^2 = \mathrm{span}\left\{e_1, e_2, e_3, e_4\right\}$ such that each is stabilized by $A$?
This question is an extension of a previously asked question, here:
Uniqueness of Stabilized Planes in $SO(n)$ ($n$ odd)?.
Thanks everybody!
Considered as acting on $\mathbb C^2$ alone, $A$ is simply the matrix
$$\left( \begin{matrix} e^{i \theta} & 0 \\ 0 & e^{-i \theta} \end{matrix}\right),$$
which has distinct eigenvalues $e^{\pm i \theta}$ and thus has a unique pair of 1-dimensional eigenspaces. Since the action is $\mathbb C$-linear, a 1-dimensional stabilized subspace is necessarily an eigenspace, so the answer to your question is yes.