"Sibling rivalry" is a weak phrase for what I've actually experienced with my kids. My brother and I had a "rivalry." My children fight to the death.
Over the years, I've tried a lot of things from "let them duke it out, eventually they'll find it wasn't worth it," to "leap to intervene asap" to "punish both kids." The trouble with options one and three is that the kids already don't like fighting, they simply don't have the knowledge or maturity to avoid it. And the trouble with option two is that I'm only one person and after the first few dozen times dragging children apart, I get exhausted.
But things have gotten a lot better recently, and I feel like I have some level of a handle on the reasons they fight and how to prevent them. Don't get me wrong, part of the reason things are better now is that my children are older. It is possible that any family with an autistic four-year-old, a two-year-old, and an infant is going to involve a lot of kicking and biting, because none of the children has the self-control not to behave that way. But I'm going to take credit for at least some of the improvement, because I notice it quickly when I implement my ideas.
Basically, my system is to stop waiting for fights to break out, and start orchestrating positive interactions ahead of time. I used to think micromanaging children's interactions was too helicoptery, but given the amount of conflict we had, it really is necessary. I am starting with the assumption that my children want to get along with each other but don't know how. So I need to be very involved, at least some of the time, to teach them directly the many skills that go into getting along with a peer.
I mean, what are the reasons for conflict among children? One of the biggest reasons is a lack of understanding. One child thinks his sister is going to push him, when she was just trying to get by. Or she thinks the comment her brother made was specifically intended to upset her, when it wasn't.
Another reason is the lack of self-control; a child quickly escalates from "someone offended me" to smacking without a whole lot of thought going into it.
A third reason is that a child is rankling over something that happened earlier, and is seeing a sibling as an enemy till further notice.
The last reason that comes to mind is that children don't know how to negotiate for compromises, so they assume that anyone who isn't doing exactly what they want is going to do nothing they want.
So, watching their interaction, I can often intervene at the point when trouble first starts to brew. For instance, Michael says "I'm a lion" and Miriam starts to shriek because she thinks this means no one is going to play princesses. I can quickly jump in and say, "Michael, Miriam wants to play princesses instead. Can this be a princess game with lions in it?" Often this is all it takes. Or, "Marko, Miriam isn't going to hit you, she was hoping she could hug you, is that okay?" Or, "Michael is pretending we're in the North Pole, he's not saying wrong things on purpose to upset you." That helps a lot.
Another thing, which I've mentioned before, is making sure kids are aware that I will intervene quickly. When I physically stop fights as soon as they start, or before the first blow falls, they stop being so hypervigilant and seeing conflict everywhere. And when I verbally scold or give a timeout to the aggressor, the victim gets over their offense a lot faster. They want to know that something has been done to ensure it won't happen again.
Last of all, and this is something that can be done at any time, is to just take over and organize positive interactions among the kids. Very often they would like to play together but can't think of a game they all like. Or they'd like to play together but don't know how to start. Or sometimes one kid is feeling crabby at another, so getting just those two together and suggesting some fun things they can do goes for a lot.
All of this is especially important in our family, because Marko is the oldest and frankly terrible at the leadership skills other oldest children often have. But it helps all the kids. They're learning, bit by bit, that if they want to play with their siblings, or they want to feel that their siblings like them, they have to start something. Sometimes Marko thinks the others don't like him, so I encourage him to do something nice for them. Sure enough, they are happy about that and do it back to him. They give each other thumbs-ups and hugs and offer to share favorite toys.
I have had this fear, ever since the boys were three and one and started fighting over everything, that my kids were going to grow up hating each other. So far, that isn't so. But it does take a heck of a lot of work to manage. I'm hoping as they grow in social skills and self-control, it will slowly take less.