We prove the following assertion. Let A and B be nonsingular unitoids with simple canonical angles. Assume that their cosquares = A–*A and = B–*B commute. Bring both cosquares to diagonal form by one and the same similarity (simultaneous diagonalization). The resulting diagonal matrices Λ and M have unimodular diagonal entries. Denote by DA and DB a pair of diagonal matrices whose cosquares are Λ and M, respectively. The congruence orbits generated by DA and DB will be denoted by and . Let and be the points of these orbits corresponding to the same transition matrix P, that is, = P*DAP, = P*DBP. Then and satisfy the relations = and = . In theory of congruences, these relations can be regarded as a kind of substitute for the conventional permutability.
