Were the Rightly Guided Caliphs chosen according to the principles of Islamic Sharia (Shura), and is it true that Ali did not pledge allegiance to Caliph Abu Bakr, and if so, why?

The Companions consulted one another regarding the succession of Abu Bakr, and they agreed to put him forward due to their knowledge of his virtue and his being put forward by the Messenger of Allah, peace and blessings be upon him, to lead the prayer. Ali pledged allegiance to him. As for Umar, Abu Bakr appointed him as his successor after consulting the Companions, which is one of the legitimate means of succession. Regarding Uthman, Abd al-Rahman consulted the Companions about him. As for Ali, the Companions consulted one another about him after Uthman's martyrdom, and the people of authority and influence in Medina agreed upon him.