general 4D rotation is split into left- and right-isoclinic factors. In quaternion language Van Elfrinkhof's formula reads or, in symbolic form, According to the German mathematician Felix Klein this formula was already known to Cayley in 1854. Quaternion multiplication is associative. Therefore, which shows that left-isoclinic and right-isoclinic rotations commute. The eigenvalues of 4D rotation matrices The four eigenvalues of a 4D rotation matrix generally occur as two conjugate pairs of complex numbers of unit magnitude. If an eigenvalue is real, it must be ±1, since a rotation leaves the magnitude of a vector unchanged. The conjugate of that eigenvalue is also unity, yielding a pair of eigenvectors which define a fixed plane, and so the rotation is simple. In quaternion notation, a proper (i.e., non-inverting) rotation in SO(4) is a proper simple rotation if and only if the real parts of the unit quaternions and are equal in magnitude and have the same sign. If they are both zero, all eigenvalues of the rotation are unity, and the rotation is the null rotation. If the real parts of and are not equal then all eigenvalues are complex, and the rotation is a double rotation. The Euler–Rodrigues formula for 3D rotations Our ordinary 3D space is conveniently treated as the subspace with coordinate system 0XYZ of the 4D space with coordinate system UXYZ. Its rotation group SO(3) is identified with the subgroup of SO(4) consisting of the matrices In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to , , , , or in quaternion representation: . The 3D rotation matrix then becomes the Euler–Rodrigues formula for 3D rotations which is the representation of the 3D rotation by its Euler–Rodrigues parameters: . The corresponding quaternion formula , where , or, in expanded form: is known as the Hamilton–Cayley formula. Hopf coordinates Rotations in 3D space are made mathematically much more tractable by the use of spherical coordinates. Any rotation in 3D can be characterized by a fixed axis of rotation and an invariant plane perpendicular to that axis. Without loss of generality, we can take the -plane as the invariant plane and the -axis as the fixed axis. Since radial distances are not affected by rotation, we can characterize a rotation by its effect on the unit sphere (2-sphere) by spherical coordinates referred to the fixed axis and invariant plane: Because , the points lie