What is done in prayer after the tashahhud

It is narrated from Sayyiduna Ibn Mas'ud (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: "When one of you recites the tashahhud in prayer, let him say: 'Allahumma salli 'ala Muhammadin wa 'ala ali Muhammad, wa barik 'ala Muhammadin wa 'ala ali Muhammad, warham Muhammadan wa ala Muhammad, kama sallayta wa barakta wa tarahhamta 'ala Ibrahima wa 'ala ali Ibrahim, innaka Hamidun Majid' (O Allah, send Your mercy upon Muhammad (peace and blessings be upon him) and upon his family, send Your blessings upon Muhammad (peace and blessings be upon him) and upon his family, and have mercy on Muhammad (peace and blessings be upon him) and his family, as You sent mercy, blessings and compassion upon Ibrahim (peace be upon him) and upon his family; indeed You alone are Praiseworthy and Most Glorious)." And there are many supporting narrations for this principle concerning the obligatory parts of prayer.

Abd al-Muhaymin bin Abbas bin Sahl al-Sa'idi narrates from his father, who narrates from his grandfather, that the Prophet (peace and blessings be upon him) used to say: "There is no prayer for one who has no ablution, there is no ablution for one who did not mention the name of Allah (say Bismillah) over it, and there is no (complete) prayer for one who did not send blessings upon the Prophet of Allah in his prayer."
This hadith is not on the condition of the two Shaykhs, because they did not record the narrations of Abd al-Muhaymin.

It is narrated from Sayyiduna Abdullah bin Mas'ud (may Allah be pleased with him) that in the first two rak'ahs (of the prayer) the Prophet (peace and blessings be upon him) would sit (for the tashahhud so briefly) as though he were sitting on heated stones. The narrator says: We asked: Until he stood up? He said: Yes, until he stood up.
Mis'ar also narrated it from Sa'd bin Ibrahim.

(A narration reported in support of the previous hadith.)
This hadith is sahih on the condition of the two Shaykhs, but they did not record it, although both of them agreed on recording, through the chain of Amr bin Murrah, the hadith of Abu Ubaydah from Abdullah (bin Mas'ud) that he was not with the Prophet (peace and blessings be upon him) on the Night of the Jinn.