But You Know It's Not Allowed: A Case for Imagine as a Lennon/McCartney Song

When Paul McCartney decided to include secret messages to John Lennon on Ram ("you took your lucky break and broke it in two" being the type of message he admits to, "I find my love awake and waiting to be / what can be done for you, she's waiting for me" the type he doesn't), he must have known how John would react. This was after Lennon Remembers, after all, and Paul was well aware that John would respond to goading viciously - and publicly.
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ETA: One more thing. John Lennon once said of the song Let It Be: Nothing to do with The Beatles. It could've been Wings. I don't know what he's thinking when he writes Let It Be. I think it was inspired by Bridge Over Troubled Waters. That's my feeling, although I have nothing to go on. I know he wanted to write a Bridge Over Troubled Waters.. In fact, Let It Be was released before Bridge Over Troubled Waters. John was mistaken.

selenak suggested to me that maybe it was John who always wanted to write a Bridge Over Troubled Waters, and he was projecting. Bridge Over Troubled Waters was released in January of 1970, when Paul had retreated to depression in Scotland. The song offers love and support to someone who is going through a hard time. "I'm on your side", the song says. "I'll take your part".

But at the end of the day, John could not do that for Paul. And I think How Do You Sleep? is the song that he wrote because he could not figure out how to write Bridge Over Troubled Water.