[pandaman]

I think I see what you mean. In an even dimensions space a -x is a rotation ( determinant = 1), while it is not in an odd dimensions. So what you described is a rotation in 4D but, indeed, not in 3D. Also, I feel, from this follows, that you can only have a rotation axis (eigenvector for value 1) in an odd dimensions space so I was wrong, assuming there is an axis for any 4D rotation. Same as in 2D there may be none. I figure 5D will act as same as 3D in this regard.

[abecedarius]

I'm sorry, my 'rotation' was wrong -- as you point out, it would admit reflections. I'm glad to uncover this embarrassing bug, but on the other hand, oops. https://en.wikipedia.org/wiki/Rotation_(mathematics) has more lore to check against after I've done enough mental debugging.