I have a similar situation, but with two boys, 4 and 7. They fight over possession of things, toys, air, anything...all the time. I am always asking other Moms - do your kids get along? And whether they're lying or not, I have never had one (not one!) say their kids don't get along at all. So even though this may not help you - you've already helped ME by sharing your story!
One thing I'll say for "us" is that you probably, like me, cultivate their individuality, and also let them be themselves rather than "forcing" them to like each other (you LOVE your little brother, don't you Johnny?). This might not help the rivalry but I think it will make them better people over the long haul...
I read the book "Siblings Without Rivalry", and one of the things it said was to respect each person's possesions. I think that backfired in our case - now they are so into "mine" they rarely share anything.
The other thing the book says is to not try to always make it "even" for your kids. This is hard to practice, but it does matter, I think. Sometimes one kid needs underwear, and the other one doesn't. You don't need to buy the other one underwear just to make it "even".
What I'm trying to do now is downplay the importance of possessions (for me and for them) and play up time together, playing outside or in the pool together, etc. That has been the only thing that has helped - a little. It sounds a little cliche, but I helped build the importance of possession in their minds and now I'm working to dissolve that importance.
The other thing that I've tried (and had some success at) is when they're saying how much they hate each other (come on, folks, you know they all say it) we all sit down together and they have to come up with at least one thing that they like or appreciate about the other person.
This has had some surprising results - my 7 year old saying that he loves that his brother has so much energy (translation: always screaming, yelling, getting physical and generally running us ragged). But my son hit it right on the head - my little one is passionate about EVERYTHING and brings great energy to that passion. Sometimes this little exercise backfires, but always we end up smiling. It's worth a try.
And this won't help you right now, but as they get older it does get a teeny tiny bit better.
I'm interested to see what the other Mammas have to say!