Adding something to a book from an unpublished paper

As many of the people that I am spamming in real life might at this point know, I am turning my coend note into a book.

I would like to add a few pages taken from a (still) unpublished paper of mine (et al.), but I don't know exactly what is the policy here: considering that

1. the material I want to add is not exactly original; it is considered folklore or at maximum it is just an alternative proof of a known fact, or it is just recomposing a puzzle of pieces scattered in the literature.
2. my coauthor agrees to see that section turned into a chapter of my book.
3. it is not clear to me/us whether, should I decide to take this path, this would compromise the possibility for the paper to be published because it contains non-inedited material.
4. the paper has been submitted, has been rejected, and will probably be resubmitted as soon as we edit it accordingly to the referee comments. It will probably be at the same time I submit the book to the publishing house.
5. the book will not suffer from me not adding these 4-5 pages; only, I believe that those two particular proofs are useful, in-topic, and particularly aesthetically clean.
6. I'd be inclined to see who comes first, and add "the present section already appears in [X, §y]" for suitable choices of $$X\in \{journal,book\}$$ and $$y\in \sum_{x : X} \{sections_x\}$$.

Considering all this, what's better to do? How free am I? Is it something I can (and should) discuss with the editor? Is this usually something regulated by the contract I signed?

Thanks!

• Sounds like something that would also be fit on Academia SE. – Wojowu Nov 22 at 11:36
• Do you need the academic magic beans that come from having another paper in addition to the book? If not, just put it in the book and leave the paper as a preprint. – David Roberts Nov 22 at 11:42
• I definitely need some more magic beans, yes – Fosco Nov 22 at 11:59
• Assuming that you're a mathematician and that you're writing mathematics (edit: yes it's category theory), the custom may depend on disciplines, so it makes sense to post here. – YCor Nov 22 at 13:27
• Hi Simon! That's exactly what I did. – Fosco Dec 11 at 15:32