I am a problem solver, and when it comes to the bickering between my two boys, I try and look for ways to the reason for the bickering. This idea I implemented years ago to eliminate a portion of the bickering between them, specifically, who goes first.
Sometimes they both want to go first, sometimes they want the other to go first, and oh boy, did the bicker about that. Trying to keep track of who went first the last time on the various tasks involved was too much for my addled brain, so I came up with the idea of the "week of firsts".
I marked it all out on the calendar, alternating weeks, and whomever's week it was, they were first for everything, good or bad, whatever. You shower first. You get to press the elevator button first. You get tucked in first. If there's an argument, consult the calendar. It is interesting how the calendar solved so many arguments....
Now that they are older, we have progressed to "Month of Firsts" because we have coordinated this with chores, which change on a monthly basis. As my kids are approaching the age of being able to sit in the front seat, I will use this for the front seat issue as well.
It has eliminated at least a portion of things they bicker about.
But don't worry, they'll find other things. :)