Equivalence of quadruples and feedback equivalence of related pairs

In this paper we look for a generalization of a property of matrix pairs, to quadruples of matrices. It is known that it is possible to analyze the feedback equivalence of matrix pairs in terms of the similarity of the square matrices belonging to sets related to the pairs. We will determine when it is possible to analyze the equivalence of matrix quadruples in terms of the feedback equivalence of matrix pairs.