Hadith No: 1
Al-Hakim al-Nishapuri - Al-Mustadrak ala al-Sahihain
Book of Knowledge of the Companions - When the Prophet was angry, no one dared to speak to him except Ali
Volume: (3) - Page Number: (130)
4705 - Abu Bakr ibn Ishaq narrated to us, saying: Bishr ibn Musa narrated to us, saying: Muhammad ibn Sa'id ibn al-Isbahani narrated to us, saying: Sharik narrated to us, and Ahmad ibn Ja'far al-Qutti'i narrated to us, saying: Abdullah ibn Ahmad ibn Hanbal narrated to me, my father narrated to us, and al-Aswad ibn Amir and Abdullah ibn Numayr narrated to both, saying: Sharik narrated to us from Abu Rabi'ah al-Ayadi from Ibn Buraydah from his father, who said: The Messenger of Allah (peace be upon him and his holy progeny) said: "Indeed, Allah commanded me to love four of my companions and informed me that He loves them." We said, "Who are they, O Messenger of Allah? We all wish to be among them." He said, "Know that Ali is one of them." Then he fell silent. Then he said, "Indeed, Ali is one of them." Then he fell silent." This is a sound hadith according to the criteria of Muslim, and neither he nor al-Bukhari recorded it.

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Reference: Al-Mustadarak ala al-Saheehain Book 3

Hadith No: 2
Al-Hakim al-Naysaburi - Al-Mustadrak ala al-Sahihayn
Book of Knowing the Companions (r) - "Lower your gazes from Fatimah until she passes by."
Volume: (3) - Page Number: (154)
4784 - Abu Bakr Muhammad ibn Ali al-Faqih al-Shashi narrated to us, saying: Abu Talib Ahmad ibn Nasr al-Hafiz narrated to us, saying: Ali ibn Sa'id ibn Bashir narrated to us, from Ubad ibn Yaqoub, who said: Muhammad ibn Ishmael ibn Rija' al-Zubaydi narrated to us, from Ishaq al-Shaybani, from Jamij ibn Umayr, who said: "I entered with my mother upon Aisha, and I heard her from behind the veil asking her about Ali." She said, "You ask me about a man who, by Allah, I do not know of anyone who was dearer to the Messenger of Allah (peace be upon him and his holy progeny) than him, nor any woman on earth who was dearer to the Messenger of Allah (peace be upon him and his holy progeny) than his wife." This is a hadith with a sound chain of narrators, and they did not include it.

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Reference: Al-Mustadarak ala al-Saheehain Book 3

Hadith No: 3
Al-Hakim al-Nisaburi - al-Mustadrak alal-Sahihain
Book of the Knowledge of the Companions (ra) - The most beloved women to the Prophet was Fatimah
Volume: (3) - Page Number: (155)
4788 - Abu al-Abbas Muhammad ibn Ya'qub narrated to us, saying: Al-Abbas ibn Muhammad al-Duri narrated to us, saying: Shadh'an al-Aswad ibn Amir narrated to us, saying: Ja'far ibn Ziyad al-Ahmad narrated to us, from Abd Allah ibn Ata from Abd Allah ibn Buraydah, from his father, who said:The most beloved women to the Messenger of Allah (peace be upon him and his holy progeny) was Fatimah,and among the men, Ali. This is a hadith with a correct chain of narration, and they did not include it in their collections.

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Reference: Al-Mustadarak ala al-Saheehain Book 3

Hadith No: 4
Al-Hakim al-Nisapuri - al-Mustadrak ala al-Sahihain
Book of the Knowledge of the Companions (r) - Supplication for Repelling Poverty and Fulfilling Debt
Volume: (3) - Page Number: (157)
4798 - Abu Bakr ibn Abi Darm narrated to me, saying: Ibrahim ibn Abdullah al-Absi narrated to us, saying: Malik ibn Isma'il al-Nahdi narrated to us, saying: Abd al-Salam ibn Harb narrated to us, from Abu al-Juhaf from Jamij ibn Umayr: "I entered with my aunt upon Aisha (r), and she was asked, 'Which of the people was most beloved to the Messenger of Allah (peace be upon him and his holy progeny)?' She said, 'Fatimah.' It was said, 'And among the men?' She replied, 'Her husband,' adding that, 'if what I knew of him was true, he was often fasting and standing in prayer.'" This is a hadith with a sahih chain of narration, and they did not include it in their collections.

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Reference: Al-Mustadarak ala al-Saheehain Book 3