A matrix-based proof of the quaternion representation theorem for   four-dimensional rotations
  To each 4x4 matrix of reals another 4x4 matrix is constructed, the so-called associate matrix. This associate matrix is shown to have rank 1 and norm 1 (considered as a 16D vector) if and only if the original matrix is a 4D rotation matrix.   This rank-1 matrix is the dyadic product of a pair of 4D unit vectors, which are determined as a pair up to their signs. The leftmost factor (the column vector) consists of the components of the left quaternion and represents the left-isoclinic part of the 4D rotation. The rightmost factor (the row vector) likewise represents the right quaternion and the right-isoclinic part of the 4D rotation.   Finally the intrinsic geometrical meaning of this matrix-based proof is established by means of the usual similarity transformations. 
Johan Ernest Mebius