Did Prophet Muhammad PBUH asked Muslims to repeat “sallallahu ‘alayhi wa sallam” (peace be upon him) when after his name? If yes, is there a narrative within the Quran or Sahih Hadith, or do Muslims roll in the hay out of affection, love and reference to Prophet Muhammad? The Answer Yes!
As already mentioned the Holy Qur’an distinctly commands the believers to send their salutations on the Prophet Muhammad (s) during this verse.
Indeed Allah and His angels bless the Prophet. O you who have faith! Invoke blessings on him (sallu ‘alayh) and invoke Peace upon him (wa sallimu) in an exceedingly worthy manner.- Qur’an, 33:56
The Prophet (s) explained a way to send him such salutations and this can be evidenced in many authentic narrations, like this hadith from Sahih al-Bukhari:
We asked Allah's Messenger (ﷺ) saying, 'O Allah's Messenger (s)! How should one (ask Allah to) send blessings on you, the members of the family (ahl al-bayt), for Allah has taught us a way to salute you (sallim)?'
He said, 'Say: O Allah! Send Your Mercy on Muhammad and on the family of Muhammad (allahumma salli ‘ala Muhammad wa Aal Muhammad), as You sent Your Mercy on Abraham and on the family of Abraham, for you're the foremost Praise-worthy, the foremost Glorious. O Allah! Send Your Blessings on Muhammad and also the family of Muhammad, as You sent your Blessings on Abraham and on the family of Abraham, for you're the foremost Praise-worthy, the foremost Glorious.' "
The Sahih al-Bukhari, Vol. 4, Book 55, Hadith 589, online right here
The mention of the Ahl al-Bayt because the Aal Muhammad (progeny of Muhammad) alongside his own name is taught by the Prophet (s) himself. In fact, there's also a narration found in certain Sunni texts that show that when asked about the way to send greetings on him, the Prophet (s) replied:
“Do not invoke ‘incomplete’ blessings on me.”
They again asked: “How should we invoke blessings on you?”
He (s) replied: “Say: Allahumma salli ‘ala Muhammad wa Aali Muhammad - O Allah! Bless Muhammad and therefore the family of Muhammad.”
For this reason all Shi’a Muslims and Sunni Muslims round the world say sallalahu ‘alayhi wa aalihi wa sallam whenever the name of the Prophet Muhammad (s) is mentioned.