The history of Sunni vs Shia split is not black and white as you think. First off, war did not start as soon as Muhammed died. It started slightly after Uthman was assassinated. Aly was unanimously selected agreed by everyone to be the next Caliph. Few months later some disagreement came up with Muwayia. Muwayia refused to obey Aly and the disagreement became worse. Both of them choose a judge to end the problem. The judge decided that neither of them would became Caliph. The civil war between both continued and Muwayia won some battles. Aly started seeing that this isn't going anywhere and surrendered. Later Aly was assassinated. The people who supported Aly were Shiaa. This is extremely simplified as the story is much more complicated than that and you would end up with a book explaining it. Note that both Sunni and Shiaa both agree that Aly was on the right, but disagree on the divinity of Muhammed's lineage. The split was more political in nature, but became more religious decades later.