Rotations in 4D and Other Dimensions

To talk about rotations in 4D etc, we need a general definition of rotation. Geometers consider a rotation to be a rigid transformation about the origin, but not a reflection. What does this imply?

1. All the real eigenvalues must be ±1 since otherwise a line in the direction of the corresponding eigenvector would have its length changed, and not be rigid.
2. Since the eigenvalues are solutions of a polynomial with real coefficients, any complex ones must occur in conjugate pairs.
3. Therefore, in 3D, there are 1 or 3 real eigenvalues, all ±1. If there are 3, then this transformation is an identity or a reflection. If there is one eigenvalue, then its eigenvector represents the axis. This is a proof that there is, in fact, an axis line in 3D.
4. In 4D, there will be 0, 2, or 4, real eigenvalues. If 0, then the rotation moves every point but the origin. One example would be a rotation in the xy plane followed by one in the zt plane. If there are 2 real eigenvalues, then there is a plane of fixed points. If 4, then this is an identity or a reflection. There never is just a fixed axis line.