Final answer:
The last common Imam of Ismaili and Ithnaasharis is Imam Jafar al-Sadiq. He is recognized as the sixth Imam by both sects before their succession paths diverged. Shah Ismail claimed to be the Mahdi and established the Safavid dynasty.
Step-by-step explanation:
The last common Imam of Ismaili and Ithnaasharis is Imam Jafar al-Sadiq. Imam Jafar al-Sadiq is recognized by both Ismaili and Ithnaashari (Twelver) Shia Muslims as the sixth Imam. After Imam Jafar, the two sects diverged in their line of succession. For the Ismaili sect, Imam Ismail ibn Jafar was appointed as the next Imam, while the Ithnaashari sect believes the successor to be Imam Musa al-Kadhim, the younger half-brother of Imam Ismail. The Ithnaasharis (Twelver Shias) believe that the legitimate caliphs in Shia Islam must be descended from Ali, Muhammad's cousin and son-in-law, and recognize the Twelve Imams, with Imam Muhammad al-Mahdi as the twelfth and final Imam who is currently in occultation. On the subject of Shah Ismail, he emerged from exile and claimed to be the Mahdi, rallying troops and eventually establishing the Safavid dynasty, which was the only Shia caliphate represented by the Fatimids. This dynasty had roots in both spiritual leadership and was also marked by military campaigns against the Ottomans and Uzbeks, solidifying the borders of modern Iran and promoting Shia Islam in the region.