Oath of allegiance and inheritance between Abu Bakr and ‘Umar on the one hand, and ‘Ali and Faatimah on the other
Question

What is the real nature of the differences that arose between ‘Ali and his wife (may Allah be pleased with them), and ‘Umar and ‘Aa’ishah (may Allah be pleased with them)?
Praise be to Allah.Firstly: 
There were no differences in belief between the Sahaabah (may Allah be pleased with them), or even in the methodology of deriving rulings from the texts, for they were the best of generations, as the most truthful one (the Prophet – blessings and peace of Allah be upon him) said, and their relationships with one another were more noble and more sublime than suggested by the lies of the Raafidis to that effect. One of the clearest proofs of that is the ties of marriage that occurred among them, and how they named their sons after the most senior and prominent among the noble Sahaabah. 
If we move on to the topic of this question, we may note that ‘Umar ibn al-Khattaab married the daughter of ‘Ali ibn Abi Taalib and Faatimah, whose name was Umm Kulthoom. Among the names of al-Husayn’s sons were Abu Bakr and ‘Umar. This is how it was, and with regard to the difference that arose between Faatimah (may Allah be pleased with her) and Abu Bakr as-Siddeeq, it was Abu Bakr (may Allah be pleased with him) who was in the right. Faatimah (may Allah be pleased with her) wanted to inherit from her father, who was the Prophet (blessings and peace of Allah be upon him), but Abu Bakr (may Allah be pleased with him) told her that the Prophets are not to be inherited from. This is what he had heard from the Prophet (blessings and peace of Allah be upon him), and Abu Bakr had no self-interest in that, for Allah, may He be exalted, had made him wealthy and independent of means. He refused to let Faatimah inherit, just as he refused to let his daughter ‘Aa’ishah – who was the wife of the Prophet (blessings and peace of Allah be upon him) inherit likewise. That was not based on self-interest on his part, and there was nothing between him and Faatimah that would generate enmity between him and her. ‘Ali (may Allah be pleased with him) stood by his wife Faatimah in order to console her following the loss of her father and comfort her with regard to her blaming Abu Bakr for his refusal to give her anything from the estate of her father (blessings and peace of Allah be upon him). 
And he refused to go to Abu Bakr (may Allah be pleased with him) and swear allegiance to him whilst Faatimah was still alive after the death of the Prophet (blessings and peace of Allah be upon him) – she lived for six months after his death – for this reason, and for another reason, which was that he thought that the Sahaabah had been too hasty in selecting the caliph and swearing allegiance to him, and that he should have been consulted, and he should have been present, although the Sahaabah thought otherwise. When Faatimah (may Allah be pleased with her) died and was buried, ‘Ali rethought the matter and asked Abu Bakr to come to his house. He came and ‘Ali explained why he had delayed swearing allegiance, then Abu Bakr insisted that he had been correct to withhold the estate of the Prophet (blessings and peace of Allah be upon him). ‘Ali was thus reassured, and he made an appointment to swear allegiance to him on the same day at Zuhr. He swore allegiance to him, and the Muslims rejoiced greatly. 
This is a summary of what happened, and this was narrated by al-Bukhaari and Muslim. ‘Ali (may Allah be pleased with him) did not refuse to obey Abu Bakr, and he did not want to cause division among the Muslims. It is not a condition the caliph’s legitimacy that every Muslim should swear allegiance to him. 
There follows the text of the hadith in full: 
It was narrated from ‘Aa‘ishah that Faatimah, the daughter of the Messenger of Allah (blessings and peace of Allah be upon him), sent word to Abu Bakr al-Siddeeq, asking for her inheritance from the Messenger of Allah (blessings and peace of Allah be upon him), from the booty that Allah had granted him in Madinah and Fadak, and what was left of the booty of Khaybar. Abu Bakr said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “We (Prophets) are not to be inherited from, and whatever we leave behind is charity. Rather the family of Muhammad may live on this wealth (that belongs to Allah).” By Allah, I will not change any of the charity of the Messenger of Allah (blessings and peace of Allah be upon him) from how it was at the time of the Messenger of Allah (blessings and peace of Allah be upon him), and I will do the same with it as the Messenger of Allah (blessings and peace of Allah be upon him) did. Abu Bakr refused to give Faatimah anything of it, and Faatimah felt angry with Abu Bakr for that, and she forsook him and did not speak to him until she died, and she lived for six months after the Messenger of Allah (blessings and peace of Allah be upon him). When she died, her husband ‘Ali ibn Abi Taalib buried her at night, and did not tell Abu Bakr about that, and ‘Ali offered the funeral prayer for her. ‘Ali was held in special esteem among the people during Faatimah’s lifetime, but when she died, ‘Ali noticed a change in the people’s attitude towards him. He sought to reconcile with Abu Bakr and swear allegiance to him, as he had not sworn allegiance to him all those months. He sent word to Abu Bakr saying: Come to us and no one else should come with you – objecting to the presence of ‘Umar. ‘Umar said: No, By Allah, you should not enter upon them alone. Abu Bakr said: What will they do to me? By Allah, I will go to them. So Abu Bakr entered upon them, and ‘Ali ibn Abi Taalib recited the Tashahhud, then he said: O Abu Bakr, we acknowledge your virtue and what Allah has given you. We do not envy you for any favour that Allah has bestowed upon you, but you did it without consulting us and we thought that we had the right (to be consulted) because of our kinship with the Messenger of Allah (blessings and peace of Allah be upon him). He kept speaking to Abu Bakr until Abu Bakr’s eyes filled with tears. When Abu Bakr spoke, he said: By the One in Whose hand is my soul, upholding kinship with the Messenger of Allah (blessings and peace of Allah be upon him) is dearer to me than upholding kinship with my own people. As for this dispute that occurred between me and you concerning these properties, I have not deviated from the right path with regard to them, and I have not given up something that I saw the Messenger of Allah (blessings and peace of Allah be upon him) do with them; rather I have done it too. ‘Ali said to Abu Bakr: Your appointment for my oath of allegiance is this afternoon. When Abu Bakr had prayed Zuhr, he ascended the minbar and recited the tashahhud, and he spoke of ‘Ali and his delay in swearing allegiance, and the excuse that he had given, then he prayed for his forgiveness. And ‘Ali ibn Abi Taalib recited the tashahhud and spoke highly of Abu Bakr, and said that what he had done was not due to jealousy of Abu Bakr, or denying the virtue that Allah had bestowed upon him, but we thought that we should have had a share in the matter, but it had been decided without consulting us, and we were upset with that. The Muslims were pleased with this and said: You have done the right thing. Then the Muslims became closer to ‘Ali, when he did the right thing. 
Narrated by al-Bukhaari (3998) and Muslim (1759) 
There follow some comments on the hadith from the scholars: 
(a) An-Nawawi (may Allah have mercy on him) said:
With regard to the delay in swearing allegiance on the part of ‘Ali (may Allah be pleased with him), ‘Ali mentioned it in this hadith and apologised to Abu Bakr (may Allah be pleased with him). Furthermore, his delay did not undermine his subsequent oath of allegiance or cast any aspersions upon him. As for the legitimacy of the caliphate, the scholars are unanimously agreed that in order for the caliphate to be legitimate, it is not essential that all the people should swear allegiance, and not even all the decision-makers. Rather it is stipulated that allegiance be sworn by those who it is possible to bring together of the scholars, leaders and prominent figures. With regard to the delay not undermining ‘Ali, that is because it is not essential for everyone to come to the leader and put his hand in his and swear allegiance to him; rather what is required, if the decision-makers choose a leader, is for everyone to accept his rule and not express any dissent or cause any division. And this is how ‘Ali (may Allah be pleased with him) was during the period before he swore allegiance: he did not express any opposition to Abu Bakr or try to cause any division, but he delayed coming to him for the reason mentioned in the hadith, and the legitimacy of the caliphate did not depend on his being present. So he did not have to be present for that purpose or for any other. Because it was not required, he did not attend, but there is no report from him to suggest that he doubted the legitimacy of the caliphate or opposed it. But he felt some reservations in his heart, so he delayed until those reservations were dispelled. The reason for those reservations was that because of his high status and virtue, and his closeness to the Prophet (blessings and peace of Allah be upon him), and so on, he thought that no decision should have been made except in consultation with him and in his presence. But the justification of Abu Bakr, ‘Umar and the other Sahaabah was clear, because they believed that hastening to conclude the oath of allegiance was in the best interests of the Muslims, and they feared that delaying it could lead to dissent and disputes that would result in great mischief. That is why they delayed burying the Prophet (blessings and peace of Allah be upon him) until the caliph had been appointed and allegiance had been sworn to him, because it was the most important matter, so that there would be no dispute concerning his burial or shrouding or washing, or the offering of the funeral prayer for him, or any other issue, when they had no one to decide on these matters. So they decided that giving precedence to the appointment of the caliph was more important. 
Sharh Muslim (12/77, 78) 
(b) What is meant by Faatimah (may Allah be pleased with her) forsaking Abu Bakr (may Allah be pleased with him) was not the kind of forsaking that is prohibited, for she was a non-mahram to him in the first place
Badr ad-Deen al-‘Ayni (may Allah have mercy on him) said: 
What is meant by her forsaking is that she avoided meeting him and did not feel at ease with the idea of doing so. It was not forsaking in the prohibited sense, which refers to refraining from greeting with salaam and the like. 
‘Umdat al-Qaari Sharh Saheeh al-Bukhaari (17/258) 
And he (may Allah have mercy on him) said: 
Al-Muhallab said:
Her forsaking him took the form of avoiding meeting him and not having any contact with him, and this does not come under the heading of forsaking that is prohibited. What is prohibited in that regard is, if they happen to meet, one of them does not greet the other with salaam. But no one narrated anything to suggest that they met and refused to greet one another. Even if they had done that, they would not be regarded as forsaking or shunning one another in the prohibited sense, unless there was clear enmity between them and clear shunning of one another. Rather all that happened is that she stayed home, and the narrator described that as forsaking or shunning.
‘Umdat al-Qaari (15/20) 
(c) Al-Haafiz Ibn Hajar (may Allah have mercy on him) said:
It is as if they (the Sahaabah) understood the reason why ‘Ali refrained from swearing allegiance to Abu Bakr so long as Faatimah was still alive, because he was focused on caring for her, tending her and trying to console and comfort her in her grief at the loss of her father (blessings and peace of Allah be upon him), and because when she got angry with Abu Bakr for his refusal to grant her request to a share of the inheritance, ‘Ali though it more appropriate to stand by her and keep away from Abu Bakr. 
Fath al-Baari (7/494) 
(d) And he (may Allah have mercy on him) said:
With regard to ‘Ali requesting Abu Bakr to come by himself, not accompanied by anyone, which was described as his “objecting to the presence of ‘Umar” , the reason for that was because of what they knew of ‘Umar’s toughness and firmness in word and deed, whereas Abu Bakr was kind and gentle. It is as if they feared that, if ‘Umar joined them, he would criticise ‘Ali, which could lead to the opposite of what they were aiming for of reconciliation. 
Fath al-Baari (7/494) 
(e) And he (may Allah have mercy on him) said:
Ali’s words “We do not envy you for any favour that Allah has bestowed upon you” mean: we do not envy you for the caliphate. With regard to his words “but you did it without consulting us”, what is referred to is his assuming the caliphate. 
(f) His words “because of our kinship with the Messenger of Allah (blessings and peace of Allah be upon him)” mean: because of our closeness to the Messenger of Allah (blessings and peace of Allah be upon him) we thought that we had a share in this matter.
Fath al-Baari (7/494) 
This is what happened between Faatimah and Abu Bakr, and between ‘Ali and Abu Bakr (may Allah be pleased with them all), and anyone who hears and reads what the Raafidis say will be shocked by their lies. Here we see ‘Ali apologising to Abu Bakr, showing respect to him, acknowledging and accepting his caliphate, and swearing allegiance to him in front of the people. Abu Bakr had previously visited him in his house and had praised him and the family of the Prophet (blessings and peace of Allah be upon him), and ‘Ali kinship to him, and he had given them precedence over his own kin. So where do you find in this story any sense of resentment, cursing, dissimulation, ill intent, evil and hatred?! None of that existed anywhere except in the minds and hearts of the Magian Shi‘ah, the allies of Hulagu and the followers of al-‘Alqami [who incited Hulagu to invade Iraq]. This report is the soundest of the reports concerning this issue, and was narrated by both al-Bukhaari and Muslim (may Allah have mercy on them). We do not conceal anything of our religion, and we have mentioned that ‘Ali did not want anyone to come with Abu Bakr, and neither ‘Aa’ishah nor anyone else hid the fact that perhaps what was meant was that he did not want ‘Umar (may Allah be pleased with him) to come. Abu Bakr agreed to that, and he thought that there was no harm in going by himself and nothing to worry about. 
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: 
Al-Qurtubi said: Whoever reflects on what happened between Abu Bakr and ‘Ali of friendly discussion, how they resolved the issue, and how they were fair and just with one another, will realise that they used to acknowledge one another’s virtue, and that there was mutual respect and love between them. Even though human nature might take over sometimes, their piety and righteousness would overcome it. And Allah is the source of strength. 
But the Raafidis hold on to the fact that ‘Ali delayed swearing allegiance to Abu Bakr until Faatimah died, and their crazy talk about that is well known. This hadith is a refutation of their arguments on this issue. 
Fath al-Baari (7/495) 
And Allah knows best.