Uniqueness of Stabilized Planes in $SO(n)$ ($n$ odd)? Let $k>0$ be even.  Denote elements of $SO(2)$ by $R_\theta$.  We know that every element, $A$, of $SO(n)$ can be conjugated into the form $\mathrm{diag}(1, R_{\theta_1}, \dots, R_{\theta_k})$.  Denote by $H_i$ ($i = 1, \dots, k$) the corresponding stabilized planes.  My question is this:
Assuming that none of the $\theta_i$ is zero, is $\left\{ H_i \right\}$ the only collection of planes that are stabilized by $A$?
This question can of course be phrased for $n$ being even as well, but I am currently working in my research with the Lie group $G_2 \subset SO(7)$, so I wanted ask the question within that context.  
Thanks everyone!

No: consider the case when the $\theta_i$ are all equal and think of $\mathbb{R}^{2k+1}$ as $ \mathbb{R} \times \mathbb{C}^k$. On the subspace $0 \times \mathbb{C}^k$, $A$ acts by multiplication by $e^{2\pi i \theta_1}$ on $\mathbb{C}^k$. So any subspace of the form $0 \times H$ where $H$ is a 1-dimensonal subspace of $\mathbb{C}^k$ is mapped to itself by $A$.
