Hadith No: 1
Al-Hakim al-Nisaburi - Al-Mustadrak ala al-Sahihayn
Book of the Knowledge of the Companions (r) - "I am the lord of the children of Adam, and Ali is the lord of the Arabs"
Volume: (3) - Page Number: (124)
4683 - Abu al-Abbas Muhammad ibn Ahmad al-Mahbubi narrated to us, saying: Muhammad ibn Mu'adh narrated to us, saying: Abu Hafs Umar ibn al-Hasan al-Rasibi narrated to us, saying: Abu 'Awaneq narrated to us, from Abu Bashar, from Sa'id ibn Jubair from Aisha (r): that the Prophet (peace be upon him and his holy progeny) said: "I am the lord of the children of Adam, and Ali is the lord of the Arabs." This is a hadith with a sound chain of narration, and neither of the two Sahih compendiums have recorded it. In its chain is Umar ibn al-Hasan, and I hope he is trustworthy. Were it not for this, I would have judged it authentic according to the criteria of the two Shaykhs, and there is a corroborating witness for it from the narration of Urwah from Aisha.

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Reference: Al-Mustadarak ala al-Saheehain Book 3

Hadith No: 2
Al-Hakim al-Nisaburi - Al-Mustadrak ala al-Sahihain
Book of Knowledge of the Companions (ra) - I am the Lord of the Children of Adam, and Ali is the Lord of the Arabs
Volume: (3) - Page Number: (124)
4684 - Abu Bakr Muhammad ibn Ja'far al-Qari informed us in Baghdad, saying: Ahmad ibn Ubayd ibn Nasih narrated to us, saying: al-Husayn ibn Alawan narrated to us, from Hisham ibn Urwah, from his father from Aisha (ra), who said: The Messenger of Allah (peace be upon him and his holy progeny) said, "Call me the Lord of the Arabs." I said, "O Messenger of Allah, are you not the Lord of the Arabs?" He replied, "I am the Lord of the Children of Adam, and Ali is the Lord of the Arabs." There is another witness for this from the hadith of Jabir (ra), who said: The Messenger of Allah (peace be upon him and his holy progeny) said, "Call me the Lord of the Arabs." Aisha (ra) said, "Are you not the Lord of the Arabs, O Messenger of Allah?" He replied, "I am the Lord of the Children of Adam, and Ali is the Lord of the Arabs."

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Reference: Al-Mustadarak ala al-Saheehain Book 3

Hadith No: 3
Al-Hakim al-Nisapouri - Al-Mustadrak 'ala al-Sahihayn
Book of Knowledge of the Companions (may Allah be pleased with them) - O Ali, you are a master in this world and a master in the Hereafter
Volume: (3) - Page Number: (128)
[The text is lengthy, so only the relevant excerpt is provided here]
4695 - Abu al-Fadl Muhammad ibn Ibrahim al-Muzakki narrated to us, saying: Ahmad ibn Salamah and al-Husayn ibn Muhammad al-Quttabani narrated to us, and Abu al-Hasan Ahmad ibn al-Khadr al-Shafi'i narrated to me, saying: Ibrahim ibn Abi Talib and Muhammad ibn Ishaq narrated to us, and Abu Abd Allah Muhammad ibn Abdullah ibn Umayya al-Qurashi also narrated to us:The Prophet (peace be upon him and his holy progeny) looked at me and said: "O Ali, you are a master in this world and a master in the Hereafter. Your friend is my friend, and my friend is the friend of Allah. Your enemy is my enemy, and my enemy is the enemy of Allah. Woe to anyone who bears you ill will after me." It is authentic according to the criteria of the two Shaykhs, and Abu al-Azhar is considered trustworthy by unanimous agreement among them. And when a trustworthy person narrates a hadith alone, it is considered sound according to its best interpretation...

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Reference: Al-Mustadarak ala al-Saheehain Book 3

Hadith No: 4
Al-Hakim al-Nisapouri - Al-Mustadrak ala al-Sahihayn
Book of Knowledge of the Companions (R) - Paradise longs for three: Ali, Ammar, and Salman
Volume: (3) - Page Number: (137)
4723 - Abu Bakr ibn Ishaq narrated to us, saying: Muhammad ibn Ayyub narrated to me, from Amr ibn al-Husayin al-Aqiili, who narrated to us from Yahya ibn al-Ala al-Razi, who said: Hilal ibn Abi Humayd narrated to us, from Abdullah ibn As'ad ibn Zurara, from his father, who said: The Messenger of Allah (peace be upon him and his holy progeny) said: "It was revealed to me regarding Ali three things: that he is the leader of the Muslims, the Imam of the pious, and the commander of those with white foreheads and shining faces." This is a hadith with a correct اسناد (chain of narration), and they did not include it in their works.

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Reference: Al-Mustadarak ala al-Saheehain Book 3