I prayed with the Prophet ﷺ, and he would give the salam to his right, "Peace be upon you and the mercy of Allah and His blessings," and to his left, "Peace be upon you and the mercy of Allah."
صليت مع النبي ﷺ فكان يسلم عن يمينه: السلام عليكم ورحمة الله وبركاته وعن شماله: السلام عليكم ورحمة الله.
Abu Dawud (997) narrated it, saying, Abdah bin Abdullah told us, Yahya bin Adam told us, Musa bin Qays Hadrami told us, from Salama bin Kuhayl, from Alqama bin Waail, from his father, and he mentioned it.Its chain of transmission is hasan because of Musa bin Qays, for there is disagreement about him, though his hadith is hasan. However, there is disagreement about whether Alqama bin Waail actually heard from his father. Alai transmitted in Jami Tahsil (537) from Ibn Main that he said, "He did not hear anything from his father." Others affirmed that he did hear from him. The one who did not hear from his father is actually Abdul-Jabbar bin Waail, not Alqama. For this reason, Abdull-Haq Ishbili declared the chain of Abu Dawud sahih in al-Ahkam al-Wusta (1/413), as did Zaylai in Nasb Raya (1/432) and Nawawi in al-Majmu (3/479), and Hafiz did likewise in Bulugh Maram, while Munziri remained silent on it in al-Mukhtasar.The matter became confused for Hafiz in al-Talkhis (1/271), where he said, "The hadith of Waail bin Hajar was narrated by Abu Dawud and Tabarani from the hadith of Abdul-Jabbar bin Waail, from his father, but he did not hear from him." This is what he said regarding Abdul-Jabbar, but it is a mistake on his part, for Abu Dawud did not narrate from Abdul-Jabbar at all. Rather, he narrated it from Alqama. So he got it right in Bulugh Maram.The one who narrated it through the route of Abdul-Jabbar bin Waail is Abu Dawud Tayalisi (1115), who said, Masudi told us, from Abdul-Jabbar bin Waail, who said, one of the members of my household told me, from my father, that he prayed with the Prophet ﷺ and gave the salam to his right and to his left. In this narration there is an unnamed man from his household, and I think he is his brother Alqama, since Abdul-Jabbar narrates from his brother Alqama, from their father. If this is confirmed, then the hadith goes back to Alqama bin Waail.The hadith has another chain of transmission. It was narrated by Abu Dawud Tayalisi (1114), Ibn Abu Shayba (1/298), Ahmad (18853), Bayhaqi (2/26), and Tabarani (22/41), all of them through the route of Shuba, who said, Amr bin Mura informed me, saying, I heard Abu Bakhtari narrating from Abdul-Rahman bin Yahsubi, from Waail bin Hajar, that he prayed with the Prophet ﷺ, and he would say the takbir when lowering himself and when rising, and he would raise his hands at the takbir, and he would give the salam to his right and to his left. This wording is from Abu Dawud Tayalisi.Abdul-Rahman bin Yahsubi was only deemed reliable by Ibn Hibban, so he is "acceptable" because he has corroborating narrators.A note. In one narration in Sahih Ibn Hibban (1993), from the hadith of Ibn Masud, it reads, "Peace be upon you and the mercy of Allah, peace be upon you and the mercy of Allah and His blessings." This is an addition from a reliable narrator, and it must therefore be accepted. All of it is sunnah, sometimes omitted and sometimes mentioned. Hafiz expresses astonishment at the statement of Ibn Salah that "this addition is not found in any of the books of hadith," in al-Talkhis (1/271), even though he himself attributed this same addition, also from the hadith of Ibn Masud, to Ibn Majah. The copy we have does not contain this addition, so perhaps it was present in the copy that he had.