Co-author illegitimately using results? Alice, Bob and Carlos are coauthors on two papers, based at different institutions. They are listed in that order, with Alice the corresponding author. Both papers have been published as preprints, and are at different stages of the review process at journals. 

Alice had most of the main ideas for one paper, Bob had most of them for the other. Alice has done the majority of the write ups. Carlos has helped with the write ups, has contributed few of the main ideas, but has significantly streamlined the proofs of the main results.

Because of their relatively high mathematical content, the papers are being targeted for publication in journals outside the "home discipline" of the authors. The intention all along, discussed, but not at much level of detail, between the co-authors, was to publish more accessible versions of the results in the home discipline of the authors. 

Carlos has announced in passing that he is about to submit a single authored paper to a journal in the home discipline that presents a fairly obvious application of the results, one that the coauthors had briefly discussed.

Alice and Bob are shocked, and feel that this is unfair use of the results, and contrary to at least the spirit of the collaboration. But they do not know what to do, especially as collaborative work is not the norm in their home discipline. 

The questions we have are: (1) Is Carlos's behavior inappropriate? and (2) What action, if any, should Alice and Bob take?
Alice and Bob should not worry about it.  Collaboration with Alice and Bob does not prevent Carlos from working on closely related research independently.  Maybe Carlos will even succeed in bringing additional attention to the collaboration with Alice and Bob.

There are two ways this could become a problem:


Alice and Bob insist on coauthorship of Carlos's paper, even though their only contribution was a brief discussion.  They very probably have not earned it.
Carlos does not attribute results from the coauthored papers correctly in his sole-authored work.  Since Alice and Bob apparently have not seen Carlos's paper, it's not time to think about this yet.