Is a 6D rotation chiral?

[PatrickPowers]

I once convinced myself that rotations were chiral in all even-dimensional spaces but I'm not sure of that. I'm no use at proving such things. Wikipedia says that in geometry a figure is chiral if and only if its symmetry group contains no orientation-reversing isometries. That's too technical for me.

[PatrickPowers]

I'm using a non-standard definition of chirality. I don't know what the standard definition of what I'm looking for is. Informally it's "if and only if in a space there are two or more types of rotation of spheres that are always distinguishable then rotation in that space is chiral." The spheres have to have no recognizable features other than their rotations.

Here in 3D a view from above the North Pole sees the Earth rotating counterclockwise. A view from above the South Pole sees the Earth rotating clockwise. So 3D is non-chiral by my definition. Or looking at a sphere you say it is rotating clockwise. Stand on your head and look at the sphere and it appears to be rotation counter-clockwise relative to you. Or consider the planet Uranus. It has no surface features and its plane of rotation is perpendicular to the ecliptic so no one can say its rotation is clockwise or counter clockwise. All odd-dimensional spaces are like this.

I once convinced myself geometrically that all even-dimensional spaces were chiral but I don't quite trust geometry and others think differently. I want an algebraic proof or a counterexample.

Here's what ChatGPT has to say about it.

You're essentially looking at the behavior of rotations in different-dimensional spaces and whether they can be consistently distinguished under all transformations (such as reflection or inversion).

From an algebraic standpoint, the key structure governing rotations is the special orthogonal group, SO(n):

In odd dimensions, SO(2k+1) has a determinant of +1, and its rotations include reflections that flip handedness (parity transformations exist within O(n), the full orthogonal group, which contains both proper and improper rotations). This allows different rotations to appear indistinguishable under reflection, making the space non-chiral.
In even dimensions, SO(2k) is fundamentally different because it has two distinct connected components in its double cover, Spin(2k), corresponding to left- and right-handed rotations. This often leads to an inherent chirality.

This supports my conclusion so naturally I like it. I'm going to have some humans check it out.

Silence. Apparently the humans don't understand this either.

[steelpillow]

It all depends on whether your space is orientable, i.e. whether you can distinguish left from right in it. This is a topological property of some n-space when treated as an n-surface or "manifold".
The easiest way to visualise it is to take some limiting "edge" to the space. If a rotation can slide some part of the figure across the limit, such that it reappears on the other side in the same orientation, then the space is orientable and rotations are chiral. But if it reappears mirror-imaged then the space is not orientable and rotations are not chiral. If the limit is "at infinity" then you get a lot of arguments on this forum.
All Euclidean spaces are orientable and have chiral rotations.
Projective spaces have the curious property than an n-space is orientable if and only if n is odd. The projective plane, 4- and 6- spaces and so on are not orientable and rotations are not chiral.
There are a huge number of 6-spaces, characterised by their Betti numbers and torsion coefficients. Some are orientable, others are not. So your answer depends entirely on which 6-space you define for yourself.

[PatrickPowers]

I presented this ChatGPT view to MathStackExchange and they said it was complete nonsense. As far as how to approach a proof, just silly comments born of cluelessness.