  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.
