My kids are fourteen and sixteen and are graduates of 1-2-3 Magic, which isn't surprising, as I have co-written two of the companion books to the original 1-2-3 Magic! I've been doing this parenting thing for quite some time and I will tell you that the behavior that my kids exhibit that drives me the most bonkers is fighting with each another. I've never been a fan of conflict. In my own life, I tend to avoid it (for better or worse!) like the plague. Since I try not to engage in battle myself, it makes me slightly nuts to listen to my kids engaging in it with one another. I can't stand it.
I have good news and bad news for you regarding sibling rivalry. Bad news first—you simply cannot eliminate it completely. Believe me, I've tried. Siblings have been going at each other since the dawn of time (Cain and Abel, anyone??). Siblings will continue to go at each other until the end of time. My two teenagers still regularly spar.
Now, having said all of that, I do have some good news for you! While you can't eliminate this behavior from your children's repertoire, you can manage it. You can keep their little spats from escalating into all out wars. I have some tips that will help you.