Please forgive any awkward phrasing or misuse of terminology. My education isn't entirely formal.
Question
Am I right in guessing these orbits trace lissajous-ish figures on hyperspheres?
Background
I'm looking at what I call "trajectories" of $\textbf{x}\in\mathbb{R}^n$ in what I call "orthogonal systems" $$\dot{\textbf{x}} = \textbf{Ax}$$ (dot meaning differential), so called because $\textbf{A}$ is orthogonal.
Let's say $\textbf{A}$ has no real only imaginary eigenvalues. Then in 2 dimensions, the orbits are a set of concentric circles. What about 4?
I think it's always possible to "block-diagonalize" $\textbf{A}$ into a form I call "pairwise-decoupled":
$$\textbf{D}^{-1}\textbf{A}\textbf{D} = \left( \begin{array}{cc} \begin{array}{cc} a_{11} && a_{12} \\ a_{21} && a_{22} \end{array} && 0 \\ 0 && \begin{array}{cc} a_{33} && a_{34} \\ a_{43} && a_{44} \end{array} \end{array} \right)$$
This seems to suggest the orbits can be decomposed into circular paths in independent planes, each with a different rate of revolution, and the decomposition is possible in any even dimensionality.
Right?