I spent the majority of my day on the phone with delta and Alitalia, but still don't have a clear answer. I arrive at JFK via Delta terminal 4 tomorrow morning. I have a separate flight with Alitalia from terminal 1 that evening. I am hoping to spend the day in the delta sky lounge. If I have checked bag what is the best way to do this? From my understanding airlines won't check your bag before 4 hours of flight boarding. Should I just pay to store luggage? Once I leave terminal 4 can I re-enter to access the sky lounge since my boarding pass will now be for Terminal 1? I appreciate any help.

I'm not sure, but wouldn't the easiest thing just taking your luggage to the Delta Lounge -- they have a store room. Then go to terminal 1 maybe 90-ish minutes before your flight.

Delta and Alitalia are both Skyteam members. Give the delta agent at your starting airport the Alitalia flight info and they should be able to tag your luggage to final destination.

Separate tickets should not matter here.

Airlines these days are loathe to check luggage through on separate tickets, even if it's the same airline

Why wouldn't you just keep the luggage with you in the Lounge for the day?

Delta is definitely not "loathe" to check luggage across ST carriers, even if on separate tickets. The only issue is the length of the layover but worst case, just stow the bag at the Sky Club.