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.'"  

طَلَبْتُ النَّبِيَّ ﷺ فَلَمْ أَقْدِرْ عَلَيْهِ فَجَلَسْتُ فَإِذَا نَفَرٌ هُوَ فِيهِمْ وَلاَ أَعْرِفُهُ وَهُوَ يُصْلِحُ بَيْنَهُمْ فَلَمَّا فَرَغَ قَامَ مَعَهُ بَعْضُهُمْ فَقَالُوا يَا رَسُولَ اللَّهِ فَلَمَّا رَأَيْتُ ذَلِكَ قُلْتُ عَلَيْكَ السَّلاَمُ يَا رَسُولَ اللَّهِ عَلَيْكَ السَّلاَمُ يَا رَسُولَ اللَّهِ عَلَيْكَ السَّلاَمُ يَا رَسُولَ اللَّهِ قَالَ إِنَّ عَلَيْكَ السَّلاَمُ تَحِيَّةُ الْمَيِّتِ إِنَّ عَلَيْكَ السَّلاَمُ تَحِيَّةُ الْمَيِّتِ ثَلاَثًا ثُمَّ أَقْبَلَ عَلَىَّ فَقَالَ إِذَا لَقِيَ الرَّجُلُ أَخَاهُ الْمُسْلِمَ فَلْيَقُلِ السَّلاَمُ عَلَيْكُمْ وَرَحْمَةُ اللَّهِ ثُمَّ رَدَّ عَلَىَّ النَّبِيُّ ﷺ قَالَ وَعَلَيْكَ وَرَحْمَةُ اللَّهِ وَعَلَيْكَ وَرَحْمَةُ اللَّهِ وَعَلَيْكَ وَرَحْمَةُ اللَّهِ  

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.  

أَتَيْتُ النَّبِيَّ ﷺ فَقُلْتُ عَلَيْكَ السَّلاَمُ فَقَالَ لاَ تَقُلْ عَلَيْكَ السَّلاَمُ وَلَكِنْ قُلِ السَّلاَمُ عَلَيْكَ وَذَكَرَ قِصَّةً طَوِيلَةً وَهَذَا حَدِيثٌ حَسَنٌ صَحِيحٌ  

"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."  

أَنَّ رَسُولَ اللَّهِ ﷺ كَانَ إِذَا سَلَّمَ سَلَّمَ ثَلاَثًا وَإِذَا تَكَلَّمَ بِكَلِمَةٍ أَعَادَهَا ثَلاَثًا