Sahih Al Bukhari: 8.813: Bk 86.53
https://sunnah.com/bukhari:6824
Narrated Ibn `Abbas: When Ma'iz bin Malik came to the Prophet (in order to confess), the Prophet said to him, "Probably you have only kissed (the lady), or winked, or looked at her?" He said, "No, O Allah's Apostle!" The Prophet said, using no euphemism, "Did you have sexual intercourse with her?" The narrator added: At that, (i.e. after his confession) the Prophet ordered that he be stoned (to death).

Sahih Al Bukhari: 8.814: Bk 86.54
https://sunnah.com/bukhari:6825
Narrated Abu Huraira: A man from among the people, came to Allah's Apostle while Allah's Apostle was sitting in the mosque, and addressed him, saying, "O Allah's Apostle! I have committed an illegal sexual intercourse." The Prophet turned his face away from him. The man came to that side to which the Prophet had turned his face, and said, "O Allah's Apostle! I have committed an illegal intercourse." The Prophet turned his face to the other side, and the man came to that side, and when he confessed four times, the Prophet called him and said, "Are you mad?" He said, "No, O Allah's Apostle!" The Prophet said, "Are you married?" He said, "Yes, O Allah's Apostle." The Prophet said (to the people), "Take him away and stone him to death." Ibn Shihab added, "I was told by one who heard Jabir, that Jabir said, 'I was among those who stoned the man, and we stoned him at the Musalla (`Id praying Place), and when the stones troubled him, he jumped quickly and ran away, but we overtook him at Al−Harra and stoned him to death (there).' "

Sahih Al Bukhari: 8.815: Bk 86.55
https://sunnah.com/bukhari:6827
Narrated Abu Huraira and Zaid bin Khalid: While we were with the Prophet , a man stood up and said (to the Prophet ), "I beseech you by Allah, that you should judge us according to Allah's Laws." Then the man's opponent who was wiser than him, got up saying (to Allah's Apostle) "Judge us according to Allah's Law and kindly allow me (to speak)." The Prophet said, "'Speak." He said, "My son was a laborer working for this man and he committed an illegal sexual intercourse with his wife, and I gave one−hundred sheep and a slave as a ransom for my son's sin. Then I asked a learned man about this case and he informed me that my son should receive one hundred lashes and be exiled for one year, and the man's wife should be stoned to death." The Prophet said, "By Him in Whose Hand my soul is, I will judge you according to the Laws of Allah. Your one−hundred sheep and the slave are to be returned to you, and your son has to receive one−hundred lashes and be exiled for one year. O Unais! Go to the wife of this man, and if she confesses, then stone her to death." Unais went to her and she confessed. He then stoned her to death.

Sahih Al Bukhari: 8.816: Bk 86.56
https://sunnah.com/bukhari:6829
Narrated Ibn `Abbas: `Umar said, "I am afraid that after a long time has passed, people may say, "We do not find the Verses of the Rajam (stoning to death) in the Holy Book," and consequently they may go astray by leaving an obligation that Allah has revealed. Lo! I confirm that the penalty of Rajam be inflicted on him who commits illegal sexual intercourse, if he is already married and the crime is proved by witnesses or pregnancy or confession." Sufyan added, "I have memorized this narration in this way." `Umar added, "Surely Allah's Apostle carried out the penalty of Rajam, and so did we after him."

Another missing verse inside the exclamations.
Sahih Al Bukhari: 8.817: Bk 86.57
https://sunnah.com/bukhari:6830
Narrated Ibn `Abbas: I used to teach (the Qur'an to) some people of the Muhajirln (emigrants), among whom there was `Abdur Rahman bin `Auf. While I was in his house at Mina, and he was with `Umar bin Al−Khattab during `Umar's last Hajj, `Abdur−Rahman came to me and said, "Would that you had seen the man who came today to the Chief of the Believers (`Umar), saying, 'O Chief of the Believers! What do you think about so−and−so who says, 'If `Umar should die, I will give the pledge of allegiance to such−and−such person, as by Allah, the pledge of allegiance to Abu Bakr was nothing but a prompt sudden action which got established afterwards.' `Umar became angry and then said, 'Allah willing, I will stand before the people tonight and warn them against those people who want to deprive the others of their rights (the question of rulership). `Abdur−Rahman said, "I said, 'O Chief of the believers! Do not do that, for the season of Hajj gathers the riff−raff and the rubble, and it will be they who will gather around you when you stand to address the people. And I am afraid that you will get up and say something, and some people will spread your statement and may not say what you have actually said and may not understand its meaning, and may interpret it incorrectly, so you should wait till you reach Medina, as it is the place of emigration and the place of Prophet's Traditions, and there you can come in touch with the learned and noble people, and tell them your ideas with confidence; and the learned people will understand your statement and put it in its proper place.' On that, `Umar said, 'By Allah! Allah willing, I will do this in the first speech I will deliver before the people in Medina." Ibn `Abbas added: We reached Medina by the end of the month of Dhul−Hijja, and when it was Friday, we went quickly (to the mosque) as soon as the sun had declined, and I saw Sa`id bin Zaid bin `Amr bin Nufail sitting at the corner of the pulpit, and I too sat close to him so that my knee was touching his knee, and after a short while `Umar bin Al−Khattab came out, and when I saw him coming towards us, I said to Sa`id bin Zaid bin `Amr bin Nufail "Today `Umar will say such a thing as he has never said since he was chosen as Caliph." Sa`id denied my statement with astonishment and said, "What thing do you expect `Umar to say the like of which he has never said before?" In the meantime, `Umar sat on the pulpit and when the callmakers for the prayer had finished their call, `Umar stood up, and having glorified and praised Allah as He deserved, he said, "Now then, I am going to tell you something which (Allah) has written for me to say. I do not know; perhaps it portends my death, so whoever understands and remembers it, must narrate it to the others wherever his mount takes him, but if somebody is afraid that he does not understand it, then it is unlawful for him to tell lies about me. Allah sent Muhammad with the Truth and revealed the Holy Book to him, and among what Allah revealed, was the Verse of the Rajam (the stoning of married person (male & female) who commits illegal sexual intercourse, and we did recite this Verse and understood and memorized it. Allah's Apostle did carry out the punishment of stoning and so did we after him. I am afraid that after a long time has passed, somebody will say, 'By Allah, we do not find the Verse of the Rajam in Allah's Book,' and thus they will go astray by leaving an obligation which Allah has revealed. And the punishment of the Rajam is to be inflicted to any married person (male & female), who commits illegal sexual intercourse, if the required evidence is available or there is conception or confession. !!!!!!!!!!!!!!!!!! And then we used to recite among the Verses in Allah's Book: 'O people! Do not claim to be the offspring of other than your fathers, as it is disbelief (unthankfulness) on your part that you claim to be the offspring of other than your real father.' !!!!!!!!!!!!!!!!!! Then Allah's Apostle said, 'Do not praise me excessively as Jesus, son of Marry was praised, but call me Allah's Slave and His Apostles.' (O people!) I have been informed that a speaker amongst you says, 'By Allah, if `Umar should die, I will give the pledge of allegiance to such−and−such person.' One should not deceive oneself by saying that the pledge of allegiance given to Abu Bakr was given suddenly and it was successful. No doubt, it was like that, but Allah saved (the people) from its evil, and there is none among you who has the qualities of Abu Bakr. Remember that whoever gives the pledge of allegiance to anybody among you without consulting the other Muslims, neither that person, nor the person to whom the pledge of allegiance was given, are to be supported, lest they both should be killed. And no doubt after the death of the Prophet we were informed that the Ansar disagreed with us and gathered in the shed of Bani Sa`da. `Ali and Zubair and whoever was with them, opposed us, while the emigrants gathered with Abu Bakr. I said to Abu Bakr, 'Let's go to these Ansari brothers of ours.' So we set out seeking them, and when we approached them, two pious men of theirs met us and informed us of the final decision of the Ansar, and said, 'O group of Muhajirin (emigrants) ! Where are you going?' We replied, 'We are going to these Ansari brothers of ours.' They said to us, 'You shouldn't go near them. Carry out whatever we have already decided.' I said, 'By Allah, we will go to them.' And so we proceeded until we reached them at the shed of Bani Sa`da. Behold! There was a man sitting amongst them and wrapped in something. I asked, 'Who is that man?' They said, 'He is Sa`d bin 'Ubada.' I asked, 'What is wrong with him?' They said, 'He is sick.' After we sat for a while, the Ansar's speaker said, 'None has the right to be worshipped but Allah,' and praising Allah as He deserved, he added, 'To proceed, we are Allah's Ansar (helpers) and the majority of the Muslim army, while you, the emigrants, are a small group and some people among you came with the intention of preventing us from practicing this matter (of caliphate) and depriving us of it.' When the speaker had finished, I intended to speak as I had prepared a speech which I liked and which I wanted to deliver in the presence of Abu Bakr, and I used to avoid provoking him. So, when I wanted to speak, Abu Bakr said, 'Wait a while.' I disliked to make him angry. So Abu Bakr himself gave a speech, and he was wiser and more patient than I. By Allah, he never missed a sentence that I liked in my own prepared speech, but he said the like of it or better than it spontaneously. After a pause he said, 'O Ansar! You deserve all (the qualities that you have attributed to yourselves, but this question (of Caliphate) is only for the Quraish as they are the best of the Arabs as regards descent and home, and I am pleased to suggest that you choose either of these two men, so take the oath of allegiance to either of them as you wish. And then Abu Bakr held my hand and Abu Ubada bin `Abdullah's hand who was sitting amongst us. I hated nothing of what he had said except that proposal, for by Allah, I would rather have my neck chopped off as expiator for a sin than become the ruler of a nation, one of whose members is Abu Bakr, unless at the time of my death my own−self suggests something I don't feel at present.' And then one of the Ansar said, 'I am the pillar on which the camel with a skin disease (eczema) rubs itself to satisfy the itching (i.e., I am a noble), and I am as a high class palm tree! O Quraish. There should be one ruler from us and one from you.' Then there was a hue and cry among the gathering and their voices rose so that I was afraid there might be great disagreement, so I said, 'O Abu Bakr! Hold your hand out.' He held his hand out and I pledged allegiance to him, and then all the emigrants gave the Pledge of allegiance and so did the Ansar afterwards. And so we became victorious over Sa`d bin Ubada (whom Al−Ansar wanted to make a ruler). One of the Ansar said, 'You have killed Sa`d bin Ubada.' I replied, 'Allah has killed Sa`d bin Ubada.' `Umar added, "By Allah, apart from the great tragedy that had happened to us (i.e. the death of the Prophet), there was no greater problem than the allegiance pledged to Abu Bakr because we were afraid that if we left the people, they might give the Pledge of allegiance after us to one of their men, in which case we would have given them our consent for something against our real wish, or would have opposed them and caused great trouble. So if any person gives the Pledge of allegiance to somebody (to become a Caliph) without consulting the other Muslims, then the one he has selected should not be granted allegiance, lest both of them should be killed."

Sahih Al Bukhari: 8.818: Bk 86.58
https://sunnah.com/bukhari:6831
Narrated Zaid bin Khalid Al−Jihani: I heard the Prophet ordering that an unmarried person guilty of illegal sexual intercourse be flogged one−hundred stripes and be exiled for one year. `Umar bin Al−Khattab also exiled such a person, and this tradition is still valid.