Bukhari

Hadith Commentary:
By "Aal" (family) is meant those people upon whom zakat is forbidden.
Your Ahl al-Bayt (household) refers to Ali and Fatimah and Hasan and Husayn (radi Allahu anhum).
By "salat" (blessings) is meant that your lineage remains in the world forever with blessing.

Hadith Commentary:
➊
In this hadith, "salat" (blessings) refers to those worldly and religious blessings which Allah, the Exalted, bestowed upon Ibrahim (alayhis salam) and his progeny.
Today, the majority of the nations of the world trace their lineage back to Ibrahim (alayhis salam).
Indeed, Allah, the Exalted, has granted these very blessings to our master Muhammad (sallallahu alayhi wa sallam), such that those who recite his testimony of faith are innumerable upon the face of the earth today, and at the time of the five daily prayers, the proclamation of his messengership is made with such grandeur that it is unparalleled among all the religious leaders of the world.

➋
In this salat (blessing), Ibrahim (alayhis salam) himself is also intended, because the word "Aal" (family) is either additional, or Ibrahim (alayhis salam) is included within "Aal", or Ibrahim (alayhis salam) is intended foremost.
The meaning is: O Allah! Just as You sent Your mercy upon Ibrahim (alayhis salam), we ask You to send Your mercy upon our Prophet Muhammad (sallallahu alayhi wa sallam) even more so.
Since the sending of blessings upon Ibrahim (alayhis salam) and his progeny is well-known, therefore, the comparison in sending mercy upon the Messenger of Allah (sallallahu alayhi wa sallam) is in terms of renown.
It is stated in the Noble Qur’an:
"O people of the house! May Allah’s mercy and His blessings be upon you." ()
In this noble verse, "Ahl al-Bayt" (people of the house) refers to the family and progeny of Ibrahim (alayhis salam), as the context indicates.
And Allah knows best.

Hadith Commentary: Imam al-Bukhari, in this chapter, has narrated two hadiths: from one, he has derived the permissibility of sending blessings (salat) upon non-prophets independently, and from the other, the permissibility of sending blessings upon non-prophets in conjunction (with the Prophet). Some scholars have also considered it permissible to say independently for non-prophets: "Allahumma salli 'alayh" (O Allah, send blessings upon him), and this also appears to be the inclination of Imam al-Bukhari. This is because the meaning of "salat" is also mercy. Thus, "Allahumma salli 'alayh" means: O Allah! Send Your mercy upon him. And in the narration of Abu Dawud and al-Nasa'i, it is stated as: "Allahumma aj'al salawataka wa rahmataka 'ala Aal Sa'd ibn 'Ubadah" (O Allah, make Your blessings and mercy upon the family of Sa'd ibn 'Ubadah). Some have also considered it permissible to first send blessings upon the Prophet (sallallahu alayhi wa sallam) and then include others, such as saying: "Allahumma salli 'ala Muhammad wa 'ala al-Hasan ibn 'Ali" (O Allah, send blessings upon Muhammad and upon al-Hasan ibn 'Ali), and this is the preferred view.

Regarding the sending of blessings (salat) specifically upon Ibrahim (alayhis salam) in the salat upon the Prophet (sallallahu alayhi wa sallam), some have discussed why it is not said: "Allahumma salli 'ala Musa" (O Allah, send blessings upon Musa). The answer given is that upon Musa (alayhis salam) was the manifestation of Majesty (tajalli jalali), while upon Ibrahim (alayhis salam) was the manifestation of Beauty (tajalli jamali). Therefore, the name of Ibrahim (alayhis salam) was given preference so that the request for the manifestation of Beauty would be made for the Prophet (sallallahu alayhi wa sallam). Another reason appears to be that the rank of Ibrahim (alayhis salam) is greater, as he is the forefather of the Prophets (jad al-anbiya). Musa (alayhis salam) does not have this status, and the lineage of the Prophet (sallallahu alayhi wa sallam) traces back to Ibrahim (alayhis salam). The elevation and intimate friendship (khullah) that Ibrahim (alayhis salam) attained in this world and the Hereafter has not been attained by anyone else. Therefore, it was appropriate to request for the Prophet (sallallahu alayhi wa sallam) a similar elevation and intimate friendship, which indeed has been granted to him, for even today, the number of those who mention his name reaches hundreds of millions throughout the world.

("Allahumma salli 'ala Muhammad wa 'ala aali Muhammad kama sallayta 'ala Ibrahim wa 'ala aali Ibrahim innaka Hamidun Majid. Allahumma barik 'ala Muhammad wa 'ala aali Muhammad kama barakta 'ala Ibrahim wa 'ala aali Ibrahim innaka Hamidun Majid.") (Ameen)

Hadith Commentary:
From this hadith, it is understood that when sending blessings (salat) upon the Messenger of Allah (sallallahu alayhi wa sallam), others may also be included in it, because the form of salat that the Messenger of Allah (sallallahu alayhi wa sallam) has instructed includes, along with him (sallallahu alayhi wa sallam), his pure wives and noble offspring as well.
And Allah knows best.

1295. Commentary:

➊ In this narration, the Imam has two teachers: Qutaybah and Harith ibn Miskin. Some words are missing in the narration of Qutaybah which have been mentioned by Harith ibn Miskin. The Imam has clarified this. By reading the words of the narration, it becomes clear that while reading from Qutaybah, a line was skipped, because after «اللَّهُمَّ صَلِّ», (كَمَا بَارَكْتَ) cannot come; rather, only «كَمَا صَلَّيْتَ» can come.

➋ In the aforementioned ahadith, the words of salat (salutation upon the Prophet) that have been mentioned have minor verbal differences, which are of no consequence. Any of these wordings may be recited; there is no harm in it. However, it should be the Salat Ibrahimiyyah, as is evident from the narrations.

Hadith Authentication: [وأخرجه البخاري 3369، ومسلم 407، من حديث مالك به]
Jurisprudential Points:
➊ In prayer, sending blessings (salat) upon the Prophet sallallahu alayhi wa sallam in the (first) tashahhud is recommended and superior, whereas in the second tashahhud it is obligatory (fard).
➋ The following salat (blessing) is also authentically reported from the Noble Prophet sallallahu alayhi wa sallam.
It is narrated from Ka'b ibn Ujrah radi Allahu anhu that the Messenger of Allah sallallahu alayhi wa sallam said: Say «اللَّهُمَّ صَلِّ عَلَى مُحَمَّدٍ وَعَلَى آلِ مُحَمَّدٍ كَمَا صَلَّيْتَ عَلَى إِبْرَاهِيمَ وَعَلَى آلِ إِبْرَاهِيمَ إِنَّكَ حَمِيدٌ مَجِيدٌ، ‏‏‏‏‏‏اللَّهُمَّ بَارِكْ عَلَى مُحَمَّدٍ وَعَلَى آلِ مُحَمَّدٍ كَمَا بَارَكْتَ عَلَى إِبْرَاهِيمَ وَعَلَى آلِ إِبْرَاهِيمَ إِنَّكَ حَمِيدٌ مَجِيدٌ» [صحيح البخاري 1/477 ح3370]
➌ Also see the previous hadith: 268
➍ For every religious issue, one should refer to the evidence.
➎ Supplicating for goodness for the purified wives of the Noble Prophet sallallahu alayhi wa sallam is a part of faith.
➏ Sayyiduna Abdullah ibn Umar radi Allahu anhuma would stand by the grave of the Prophet sallallahu alayhi wa sallam and send blessings upon the Prophet sallallahu alayhi wa sallam, Abu Bakr, and Umar (radi Allahu anhuma). [الموطأ 1/166 ح398 وسنده صحيح]
◄ It is narrated from Imam Nafi' that Ibn Umar radi Allahu anhuma, when returning from travel, would enter the (Prophet's) Mosque, then come to the grave (of the Prophet sallallahu alayhi wa sallam) and say: «اَلسَّلَامُ عَلَيْكَ يَا رَسُوْلَ اللهِ ! اَلسَّلَامُ عَلَيْكَ يَا أَبَا بَكْرٍ ! اَلسَّلَامُ عَلَيْكَ يَا أَبَتَاهُ !» [السنن الكبريٰ للبيهقي 5/245 وسنده صحيح، طبقات ابن سعد 4/156، وسنده صحيح]
➐ Saying salam (greeting of peace) consists of supplicatory words in which the deceased may be addressed, but except for a few specific cases, it is not established that the deceased hears. If he were hearing, then responding to the salam would be necessary, but there is no authentic hadith proving the response to salam. The narration in Sunan Abi Dawud regarding the reply to salam is weak due to being defective.
➑ Reciting any authentically established salat (blessing) is meritorious and Sunnah, but it should be clear that fabricated and self-invented forms of salat must be avoided.
➒ There is no authentic evidence in any book of hadith for forms such as Durood Lakhi, Durood Tanjeena, and Durood Taj, etc.