Rotations in 4-dimensional Euclidean space
Generating 4D rotation matrices

Four-dimensional rotations can be derived from Rodrigues' rotation formula and the Cayley formula. Let A be a 4 × 4 skew-symmetric matrix. The skew-symmetric matrix A can be uniquely decomposed as

into two skew-symmetric matrices A and A satisfying the properties A A = 0, A = − A and A = − A, where ∓ θ i and ∓ θ i are the eigenvalues of A. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices A and A by Rodrigues' rotation formula and the Cayley formula.

Let A be a 4 × 4 nonzero skew-symmetric matrix with the set of eigenvalues

Then A can be decomposed as

where A and A are skew-symmetric matrices satisfying the properties

Moreover, the skew-symmetric matrices A and A are uniquely obtained as

and

Then,

is a rotation matrix in E, which is generated by Rodrigues' rotation formula, with the set of eigenvalues

Also,

is a rotation matrix in E, which is generated by Cayley's rotation formula, such that the set of eigenvalues of R is,

The generating rotation matrix can be classified with respect to the values θ and θ as follows:
detectable_content:postscript, detectable_format:constrained_response