Quoted from Here, There and Everywhere: p.284-285

"Qkay, lads, I'm ready. Time to let yer hair down and do some rock 'n' roll." It was nearly three weeks into July, and two weeks after the prodigal son had returned to the studio when we finally heard those words from a fully recovered John Lennon. He was about to run the Beatles through a new song - the second of his to be recorded for Abbey Road (they'd done some work on "I Want You (She's So Heavy)" before the summer sessions had commenced) - and we all waited with bated breath to see what he'd come up with. "Come Together" may not have been a masterpiece, but it was a catchy, hooky tune, and even though it clearly owed a lot to Chuck Berry, its abstract, somewhat risqué lyric had that distinctive Lennon stamp. The first time he played it for us, chugging away on his acoustic guitar, it was a lot faster than the final version that made it to the album. It was Paul who suggested it be done at a slower tempo, with a "swampy" kind of sound, and Lennon went along with it uncomplainingly; he always took well to constructive criticism.

John was in a pretty good mood that day, too - he seemed to come to life when we were working on one of his own songs, rather than one of Paul's or George's. True, all three of them exhibited a lack of patience if it wasn't their song - there was always a definite drop-off in interest whenever any one of them was working on another Beatle's composition - but John was consistently the most flagrant offender.
Despite Lennon's improved frame of mind, there were clearly still underlying tensions and old wounds that hadn't healed. The band kept breaking into long, pointless jam sessions, as they had done frequently during the White Album sessions, and I could see that John was treating Paul in an off-hand manner, despite the fact that Paul came up with the electric piano lick and swooping bass line that pretty well define "Come Together." John even made a point of playing the piano line, once he'd looked over Paul's shoulder and learned the part. That would have never happened in the old days: both men knew that Paul was the better piano player, and he normally would be manning the keyboards even if they were recording a Lennon song.

John not only sang the lead, but also did all the backing vocals on "Come Together" by himself. He didn't ask either Paul or George to join in, and neither of them volunteered. Harrison didn't seem to care one way or the other, but I could see that it was getting to Paul. Finally, in some frustration, he blurted out, "What do you want me to do on this track, John?"
John's reply was a diffident "Don't worry, I'll do the overdubs on this." Paul looked a bit hurt, then angry. For a moment I thought there was going to be an explosion. Instead, he contained himself, shrugged his shoulders, and simply walked out of the studio - one of the few times he ever left a session early. Paul had to have felt humiliated, but rather than having a fight or an argument about it, he chose to just get up and leave, without any dramatics. The next day, he returned, and nothing further was ever said about it.
We happened to be working on "Come Together" just as Apollo 11 was about to land on the moon, and I raced home late that night after the session ended, so I could watch Neil Armstrong's historic first step on my newly acquired color TV. To my disappointment, the broadcast from the moon was in black and white.