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In $4$-dimensional Euclidean space, one must take account of there being either one unique (simple), or two unique (double), or two but not unique (isoclinic) planes associated with any one rotation.

Does it work the same way for general rotations in Minkowski space (general as in, rotations not necessarily confined to the spatial dimensions)? I'm having trouble finding mathematical references for rotations in Minkowski space -- does the non-Euclidean metric change anything?

I'm looking for an explanation and possibly a worked example. Thanks in advance.

user226966
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Maybe the material in this text from Miguel Javaloyes and Miguel Sánchez, from pages $20$ to $26$ is helpful. The discussion there is about eigenvalues of Lorentz transformations and invariant subspaces of $\Bbb L^n$. In particular, there is a proof of the following fact, in page $24$: if $\Lambda \in \mathrm{O}_1(n, \Bbb R)$ and $U \leq \Bbb L^n$ is $\Lambda$-invariant, then $U^\perp$ is also $\Lambda$-invariant.

Follows from this that if you have a two-dimensional plane in $\Bbb L^4$ which is invariant, and said plane is not lightlike, then its complement is also a two-dimensional plane, which will be invariant. I am not familiar with the concept of isoclinic planes, but perhaps you can figure what you need from there.

As a sidenote, in this question I tried to gather references in Lorentzian Geometry, you might find something useful there too (and if you know some other good reference, feel free to add it there).

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Ivo Terek
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