The Sunni narration dealing with Imam Ali’s bayah is found in Sahih Bukhari, the most revered book of the Sunnis:
Sahih Bukhari
Volume 5, Book 59, Number 546
She remained alive for six months after the death of the Prophet. When she died, her husband 'Ali, buried her at night without informing Abu Bakr and he said the funeral prayer by himself. When Fatima was alive, the people used to respect 'Ali much, but after her death, 'Ali noticed a change in the people's attitude towards him. So Ali sought reconciliation with Abu Bakr and gave him an oath of allegiance. 'Ali had not given the oath of allegiance during those months (i.e. the period between the Prophet's death and Fatima's death). '
Ali sent someone to Abu Bakr saying, "Come to us, but let nobody come with you," as he disliked that 'Umar should come, 'Umar said (to Abu Bakr), "No, by Allah, you shall not enter upon them alone " Abu Bakr said, "What do you think they will do to me? By Allah, I will go to them' So Abu Bakr entered upon them, and then 'Ali uttered Tashah-hud and said (to Abu Bakr), "We know well your superiority and what Allah has given you, and we are not jealous of the good what Allah has bestowed upon you, but you did not consult us in the question of the rule and we thought that we have got a right in it because of our near relationship to Allah's Apostle.”
Thereupon Abu Bakr's eyes flowed with tears. And when Abu Bakr spoke, he said, "By Him in Whose Hand my soul is to keep good relations with the relatives of Allah's Apostle is dearer to me than to keep good relations with my own relatives. But as for the trouble which arose between me and you about his property, I will do my best to spend it according to what is good, and will not leave any rule or regulation which I saw Allah's Apostle following, in disposing of it, but I will follow." On that 'Ali said to Abu Bakr, "I promise to give you the oath of allegiance in this after noon." So when Abu Bakr had offered the Zuhr prayer, he ascended the pulpit and uttered the Tashah-hud and then mentioned the story of 'Ali and his failure to give the oath of allegiance, and excused him, accepting what excuses he had offered;
Then 'Ali (got up) and praying (to Allah) for forgiveness, he uttered Tashah-hud, praised Abu Bakr's right, and said, that he had not done what he had done because of jealousy of Abu Bakr or as a protest of that Allah had favored him with. 'Ali added, "But we used to consider that we too had some right in this affair (of rulership) and that he (i.e. Abu Bakr) did not consult us in this matter, and therefore caused us to feel sorry." On that all the Muslims became happy and said, "You have done the right thing." The Muslims then became friendly with 'Ali as he returned to what the people had done (i.e. giving the oath of allegiance to Abu Bakr).