´Narrated Abu Tamimah Al-Hujaimi:` from a man among his people, who said: "I went looking for the Prophet (ﷺ) but I was not able to find him. So I sat down, and then I saw a group of people, and he was among them, but I did not recognize him. He was settling some matter between them so when he was finished, some of them stood up with him and they were saying: 'O Messenger of Allah.' When I saw that, I said: "'Alaikas-Salam (upon you be peace) O Messenger of Allah! 'Alaikas-Salam (upon you be peace) O Messenger of Allah! 'Alaikas-Salam (upon you be peace) O Messenger of Allah!' He replied: 'Indeed "'Alaikas-Salam (upon you be peace)" is the greeting for the dead.' Then he came toward me and said: 'When a man meets his Muslim brother then he should say: "As-Salamu 'Alaikum Wa Rahmatullahi Wa Barakatuh (peace be upon you, and the mercy and blessings of Allah)." Then the Prophet (ﷺ) responded to my greeting, he said: 'And may Allah's mercy be upon you, and may Allah's mercy be upon you, and may Allah's mercy be upon you.'"
Hadith Referenceسنن ترمذي / كتاب الاستئذان والآداب عن رسول الله صلى الله عليه وسلم / 2721
´Narrated Abu Ghifar Al-Muthanna bin Sa'eed At-Ta'i:` from Abu Tamimah al-Hujaimi from Jabir bin Sulaim who said: "I went to the Prophet (ﷺ) and I said: ''Alaikas-Salam (upon you be peace)' so he replied: 'Do not say "'Alaikas-Salam" rather say As-Salamu Alaik.'" And he mentioned the story in its entirety.
Hadith Referenceسنن ترمذي / كتاب الاستئذان والآداب عن رسول الله صلى الله عليه وسلم / 2722
´Narrated Anas bin Malik:` "When the Messenger of Allah (ﷺ) would give the Salam he would do so three times, and when he would say a statement, he would say it three times."
Hadith Referenceسنن ترمذي / كتاب الاستئذان والآداب عن رسول الله صلى الله عليه وسلم / 2723
Hadith Gradingالألبانی:حسن صحيح مختصر الشمائل (192)