Final answer:
The two splits in Shia history after Imam Mustansirbillah were Nizari and Mustaali sects, resulting from a succession dispute between his sons. The Fatimids established the only Shia caliphate, and Shia legitimacy is tied to descent from Ali. Shah Ismail of the Safavid Empire made Twelver Shi'ism the state religion.
Step-by-step explanation:
The two splits in Shia history after Imam Mustansirbillah were c) Nizari and Mustaali. These splits occurred after the death of Fatimid Caliph-Imam al-Mustansir Billah in 1094, leading to a succession dispute between his two sons. The Ismaili community was divided into the Nizari and Mustaali sects over this dispute. While the Nizari sect believed that Nizar, the elder son of Mustansirbillah, was the rightful Imam, the Mustaali sect supported his younger son, Mustali.
During the medieval period and beyond, this division became a defining moment in Shia history, with the Nizari Ismailis establishing their own line of Imams which continues to this day under the Aga Khan, while the Mustaali Ismailis also continued through various lineages. These events all fall under the larger historical context of the Shia and Sunni branches of Islam, including the earlier split that occurred following the death of the Prophet Muhammad, where the Shia believed that leadership should have passed directly to Ali and his descendants.
The Fatimids established the only Shia caliphate, acting as a major center of Shi'ism during their rule. The caliphs in Shia Islam were believed to be legitimate only if they were descended from Ali, the cousin and son-in-law of Prophet Muhammad, which highlights the importance of lineage in Shia Islamic doctrine. Furthermore, during the reign of Shah Ismail of the Safavid Empire, Twelver Shi'ism was proclaimed as the state religion which was a critical moment in the history of Shia Islam.