Rotations are a rigid transformation.
If we have a rigid transformation, the eigenvalues, including possible complex values, must be |z| = 1, If they are not, then we have some kind of compression / dilation, and the transformation is not rigid.
In R^3 the characteristic polynomial to determine eigenvalues is a cubic polynomial. Which means that it must have one real root.
and since |z| = 1 it must be 1 or -1.
If the product of the eigenvalues = -1, then the sphere is flipped inside-out.
and since the characteristic polynomial has real coefficients, complex eigenvalues are in conjugate pairs.
that means the eigenvalues are (1, e^it, e^-it) which is equivalent to (1, cos t + i sin t, cos t - i sin t), and could include (1,1,1) and (1,-1,-1).
What does all this mean? There is always some axis of rotation which is the eigenvector associated with the eigenvalue 1, and then there is the degree of rotation, t.