Triple metamorphosis of twofold triple systems

In a simple twofold triple system (X,B), any two distinct triples T1, T2 with |T1∩T2|=2 form a matched pair. Let F be a pairing of the triples of B into matched pairs (if possible). Let D be the collection of double edges belonging to the matched pairs in F, and let F∗ be the collection of 4-cycles obtained by removing the double edges from the matched pairs in F. If the edges belonging to D can be assembled into a collection of 4-cycles D∗, …