Both parties are to sit before the judge

It is narrated from Sayyiduna Abdullah bin al-Zubayr (may Allah be pleased with him) that there was a dispute between him and his brother Amr bin al-Zubayr. Abdullah bin al-Zubayr went in to Sa'id bin al-As while Amr bin al-Zubayr was sitting with him on the couch. Sa'id said to Abdullah: Come here (onto the couch). He said: No; rather, the judgement of the Messenger of Allah (peace and blessings be upon him) and his (peace and blessings be upon him) Sunnah is that both litigants sit (equally) before the judge.
This hadith is sahih al-isnad (authentic in its chain), but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyiduna Abdullah bin Mas'ud (may Allah be pleased with him) that whoever is faced with a judgement should judge according to the Book of Allah. If a matter comes to him that is not in the Book of Allah, he should judge according to what His Prophet (peace and blessings be upon him) judged. If a matter comes to him that is neither in the Book of Allah nor judged by His Prophet (peace and blessings be upon him), he should judge according to what the righteous have said. And if a matter comes to him that is neither in the Book of Allah, nor judged by the Prophet (peace and blessings be upon him), nor by the righteous, he should exercise ijtihad with his own opinion. And if he cannot understand the right answer, he should admit it (rather than judge) and not be ashamed.
This hadith is sahih al-isnad (authentic in its chain), but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyiduna Abu Musa al-Ash'ari (may Allah be pleased with him) that two men laid claim before the Prophet (peace and blessings be upon him) to a camel or an animal, and neither of them had any witness, so the Prophet (peace and blessings be upon him) divided that animal between the two of them (half and half).
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it.

It is narrated from Sayyiduna Abu Musa al-Ash'ari (may Allah be pleased with him) that two men laid claim to a camel, and each of them produced two witnesses, so the Prophet (peace and blessings be upon him) divided that camel between the two of them.
This hadith too is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it.

It is narrated from the Mother of the Believers, Sayyidah Umm Salamah (may Allah be pleased with her), that two men came to the Prophet (peace and blessings be upon him) who were arguing with one another over a matter of inheritance, and they had no witnesses. The Prophet (peace and blessings be upon him) ordered them to act fairly (in the division), seek what is right, then draw lots, and each of them should forgive the other.
This hadith is sahih al-isnad (authentic in its chain), but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from the Mother of the Believers, Sayyidah Umm Salamah (may Allah be pleased with her), that I was with the Prophet (peace and blessings be upon him) when two men came disputing over a matter of inheritance, and neither of them had a witness. Each of them said about his companion: O Messenger of Allah! This right of mine which I claimed belongs to so-and-so. He (peace and blessings be upon him) said: "No; rather, go, both of you, and seek what is right (in the division), then draw lots, then divide it, and let each of you forgive his companion."
This hadith is sahih (authentic) on the condition of Imam Muslim, but the two Shaykhs did not record it.

It is narrated from Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) that a man claimed a right against another, so the two of them brought their case to the Prophet (peace and blessings be upon him). He (peace and blessings be upon him) asked the claimant for witnesses, and he said: I have no witness. He (peace and blessings be upon him) said to the other man: "Take an oath." He took the oath, saying: By Allah! He has nothing owed to him by me. Then the Messenger of Allah (peace and blessings be upon him) said: "No (it is not so); rather, his right is indeed owed by you, so give him his right." Then the Messenger of Allah (peace and blessings be upon him) said to him (the one who swore): "Your testimony that there is no true deity except Allah is an expiation for your (false) oath (that is, by the blessing of your Islam Allah has settled the matter)."
This hadith is sahih al-isnad (authentic in its chain), but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyiduna Abdullah bin Mas'ud (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: "When you see that my Ummah has become afraid (of speaking the truth) and cannot say to the oppressor, 'O oppressor,' then know that they have been bid farewell (that is, they have been deprived of Allah's help and support)."
This hadith is sahih al-isnad (authentic in its chain), but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyiduna Zayd bin Arqam (may Allah be pleased with him) that the Prophet (peace and blessings be upon him) sent Sayyiduna Ali (may Allah be pleased with him) to Yemen. There, three men brought him a dispute concerning a child, each of them claiming that he was his son. The narrator says that Sayyiduna Ali (may Allah be pleased with him) called them two at a time, in turn, in private and asked: Will you willingly give up the child in favour of this third companion? They all refused. Then he said: "I see you as partners who are at odds with one another (shuraka' mutashakisun), so now I will cast lots between you." So he cast lots, and it fell to one of them; he gave the child to him and ordered him to pay the other two companions two-thirds of the blood money (diyah) (for the child). The narrator says that when this incident was mentioned to the Prophet (peace and blessings be upon him), he (peace and blessings be upon him) laughed so much that his (peace and blessings be upon him) blessed molars became visible.
The two Shaykhs turned away from al-Ajlah bin Abdullah al-Kindi altogether, although his narrations are not abandoned (matruk); the objection raised against him is on account of his particular views (madhhab).

It is narrated from Sayyiduna Abdullah bin al-Zubayr (may Allah be pleased with them both) that Zam'ah had a slave woman with whom he had relations, and it was also suspected about this slave woman that another man used to go to her. When Zam'ah died, she was pregnant, and then she gave birth to a boy who closely resembled that other suspected man. Sayyidah Sawdah (may Allah be pleased with her) mentioned this to the Prophet (peace and blessings be upon him), and he (peace and blessings be upon him) said: "As for the inheritance, it is the right of this child (because he belongs to the owner of the bed); but you, O Sawdah, observe hijab from him, for (legally) he is not your brother."
This hadith is sahih al-isnad (authentic in its chain), but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyiduna Abu Hurayrah (may Allah be pleased with him): Abu Maymunah says that I was sitting with Abu Hurayrah (may Allah be pleased with him) when a Persian woman came to him with her son; her husband had divorced her. Speaking in her Persian language, she said: "O Abu Hurayrah! My husband wants to snatch my son away from me and take him." Meanwhile her husband also came and said: "Who will dispute with me concerning my child?" Abu Hurayrah (may Allah be pleased with him) said: "I say nothing about this from myself except what I myself heard: that a woman came to the Prophet (peace and blessings be upon him) while I was sitting there, and she said: 'O Messenger of Allah! May my father and mother be ransomed for you, my husband wants to take my son away, although he benefits me by bringing me water from the well of Abu Inabah.'" He (peace and blessings be upon him) said: "Draw lots, both of you, over him." The husband said: O Messenger of Allah! Who can dispute with me concerning my son? Then the Prophet (peace and blessings be upon him) said to the boy: "O boy! This is your father and this is your mother, so take the hand of whichever of them you wish." So the boy took hold of his mother's hand, and she went away with him.
This hadith is sahih al-isnad (authentic in its chain), but the two Shaykhs (Bukhari and Muslim) did not record it.