The Rightly Guided Four Caliphs

Abu Bakr
After Muhammad's death, succession occurred in a manner customary in Arabia at the time, following a familial format. Umar, one of the Prophet's companions, proclaimed Muhammad's father-in-law, Abu Bakr, as his successor and caliph. Political considerations played a decisive role in this decision.
The people of Mecca and Medina supported the influential Qurayshite leaders. Influence, age, and a stable background tipped the balance in favor of the elder Abu Bakr over Ali, the Prophet's son-in-law. Abu Bakr's rule lasted two years (632–634), during which he not only subdued rebellious tribes but also initiated the Muslim conquests beyond the borders of Arabia. In 634, the Muslims defeated the Byzantines at Ajnadayn. Even after Abu Bakr's death, the wave of victories continued. He had appointed the popular and energetic Umar ibn al-Khattab as his successor.

Umar
Umar became the organizer of the great Arab victories. The Muslims triumphed against the Byzantines at Damascus and Yarmouk, captured Jerusalem, and conquered Iraq, Persia, Palestine, Syria, and Egypt. In 637, at the imperial council in Jabiya, Umar introduced a noble imperial constitution for the state.
In 644, Umar fell victim to the personal vengeance of a Persian slave. The reins of government passed to the Umayyad Uthman, who was proclaimed caliph by a council of the Prophet's prominent companions.

Uthman
Uthman was not a strong personality. He distributed official positions to his relatives, leading to discontent among excluded tribes. Under his initiative, the definitive version of the Quran was compiled in 653, setting the text that remains binding to this day. The Muslim victories continued during his reign. The central authority contributed little to these successes; provincial governors led the campaigns.
The first Arab fleet was established by the commander of the Egyptian army in 645. Later, rebellious members of the Egyptian army stormed Medina and killed Uthman in 656. In Mecca, opponents of the Umayyads proclaimed Ali, the Prophet's cousin and son-in-law, as caliph.

Ali
Ali's marriage to Fatimah, Muhammad's daughter, and the fact that his sons (Hasan and Husayn) were the Prophet's only bloodline descendants, may have influenced his selection. Crucially, Ali belonged to the Hashimites, Muhammad's clan, and was part of the Prophet's closest circle of confidants.
As a scholar of tradition (sunna), Ali earned great respect, bolstered by his personal integrity. Opposing the Umayyad aristocracy, he aligned more closely with city dwellers and Bedouins. However, Ali was neither a skilled diplomat nor a successful military leader, though he stood out for his personal courage.
Ali faced a difficult position: most governors and military commanders sided with the Umayyad faction. Muawiya, ruler of Syria, and the aristocracy of the holy cities refused to submit to him. Ali managed to quell a rebellion in Basra, but Muawiya presented himself as the avenger of Uthman, whose killers had found refuge in Ali's circle.
In 657, Ali and Muawiya clashed at the Battle of Siffin. By 660, Muawiya declared himself caliph. In 661, Ali fell victim to a Kharijite assassination.
The struggle was decided; the era of Arab state formation concluded, and the Umayyad Empire began.

Burchard Brentjes "Izmael fiai", Kossuth könyvkiadó, Budapest, 1986