So here is how you do it. When two children start to fight, you tell them, "It sounds like it's time for the "Fix-it" chairs," and then you sit each one in a chair that is near enough so they can talk, but not so near they can touch each other. You give them time to calm down, sometimes I will set a timer for a couple of minutes or I just say, when you are calmed down and ready to talk to each other, let me know.
The next step is to ask each child why the OTHER child would be sad. They are not allowed to say why they are mad, only why their sibling is upset. Then you ask the sibling if they got it right. If not, let the sibling explain, and then see if the other child can repeat in their own words what was said. You can't move on until the opposite person is satisfied that their pain is understood.
After one child does this, then it is the other child's turn. Repeat the same thing you did in the previous paragraph. This is to help the child stop focusing on themselves so that they are getting more geared up to compromise. But they still get a chance to be heard because the other person has to keep saying why the opposite person is upset until they get it right.
When each child is satisfied that their problem has been stated correctly, THEN you go to the problem solving stage. You say, "Jimmy has this problem and you have that problem, can you come up with ideas to make both people happy?" Then they talk it out until they get a solution. Usually they can come up with one themselves, although at times, I have made tiny suggestions. But the goal is that you want them to be able to do it themselves so they can do it at school when they have to solve their own conflicts.
If they start to get upset again, I give them more time to calm down by saying, "Uh oh. It sounds like we haven't calmed down all the way. I'll just go away until you guys are calmed down and ready to talk." Usually that makes them calm down fast and keep working on solutions.
When they've come up with a solution, they can leave. This method makes everybody happy. It's true you have to spend time teaching them how to do it, but after they get the hang of it, they can pretty much do it themselves.
And one more thing, this book is also a GREAT resource on dealing with sibling rivalry. I have used many suggestions from it as well.
And one last word of advice to make your life easier. NEVER compare your children to each other in any way. This will really increase the rivalry in your home! Good luck!