I went to a training seminar yesterday and it included watching a short video called, "Supporting Children in Resolving Conflicts." I would definitely recommend renting/buying this to watch it. It is really helpful to see this being done and to watch someone actually try this technique. (It was with pre-schoolers, but you could adapt it for your kids).
Anyway, it includes approaching the situation calmly--don't let your emotions get to you because that can translate onto the kids' reaction. I know, easier said than done.
-Next, ask questions to find out what the problem is. Repeat what each says to make sure there is an understanding. This shows them that you understand, and when they know someone understands, often right away some of the anger subsides. (re-phrase hurtful words for them when you are repeating).
-Then ask each one how they might solve the problem. As the adult, it's important for you to accept the response/idea for how to solve the problem. For example, in the video, one child did not want another child playing with him so they were fighting. His solution was that the other child play somewhere else. She had to accept that, but then she stayed with the other child and helped him to find his own solution. She helped him get involved in a different activity so that he was not upset.
Most teachers/parents who hear about this problem solving strategy say, "I don't have time!" But, as the presenter yesterday pointed out, every minute you spend teaching your children how to problem solve on their own, is a minute saved later on. This can take time and practice for both you and the kids. It's a learning process. Think of how good you'll feel the first time you over hear them solving something on their own without you. (They may be in their 20's, but ya know). This is a skill they will use as adults too. It's so easy to just jump right into the middle of a conflict, but teaching them how to do it on their own, step by step, will give them skills they will use for a lifetime. Just think of that phrase, "If you give a man a fish, you feed him for a day. If you teach him how to fish, you feed him for life." Same concept here.
I also wanted to add that my sister and I (one day less than a year apart in age) fought ALL THE TIME growing up. We are now best friends. It didn't happen until we went to college though. I think it can just take some growing up (and maybe some distance, too).
Oh, you might also want to make up some games where they have to work together as a team. They could be on a team against the grown-ups where they have to work together. If they start to fight, again, walk them through the problem solving.
Hope this helps a little.