Add to this the Doctrine of the Concensus, which states that whatever the majority of the faithful believe to be true, is true, and you have a mess to deal with.
This description is wrong. Not the majority decides, but the consensus must be unanimous (hadith: My community will never agree in an error).
Consensus (ijma') is the least root of jurisprudence (usul al-fiqh); the first ones being Koran, Hadith (reports of sayings and doings of Muhammad), qiyas (conclusion by analogy).
The split started with a death and has led to many more. The split started before the election of Ali Ibn-Abi-Talib as 4. successor (khalifa) of Muhammad. The shi'a (=party) were the followers of Ali who saw him as the only Muslim fit for khalifa because he was cousin, foster brother, and son in law of the prophet. Furthermore he was considered by them the first follower of Muhammad. Only through his sons the bloodline of the prophet was continued, and so the shi'a always supported his descendants.