"I heard Jabir bin Samurah say: 'When we prayed behind the Prophet ﷺ we used to say: As-salamu 'alaykum, as-salamu 'alaykum (peace be upon you, peace be upon you)" - and Mis'ar (one of the narrators) pointed with his hand to the right and the left. He ﷺ said: 'What is the matter with these people who wave their hands as if they are the tails of wild horses? It is sufficient for one to place his hands on his thighs and to say the salam to his brother to his right and left." (Using translation from Nasāʾī 1318)   

كُنَّا إِذَا صَلَّيْنَا خَلْفَ النَّبِيِّ ﷺ قُلْنَا بِأَيْدِينَا السَّلَامُ عَلَيْكُمْ السَّلَامُ عَلَيْكُمْ فَقَالَ رَسُولُ اللهِ ﷺ مَا بَالُ هَؤُلَاءِ الَّذِينَ يَرْمُونَ بِأَيْدِيهِمْ كَأَنَّهَا أَذْنَابُ الْخَيْلِ الشُّمْسِ أَمَا يَكْفِي أَحَدَكُمْ أَنْ يَضَعَ يَدَهُ عَلَى فَخِذِهِ ثُمَّ يُسَلِّمَ عَلَى أَخِيهِ عَنْ يَمِينِهِ وَشِمَالِهِ  

The Prophet ﷺ commanded us to respond to the salutation of the imam. and to love each other, and to salute each other. (Using translation from Abū Dāʾūd 1001)   

أَمَرَنَا النَّبِيُّ ﷺ أَنْ نَرُدَّ عَلَى الْإِمَامِ وَأَنْ نَتَحَابَّ وَأَنْ يُسَلِّمَ بَعْضُنَا عَلَى بَعْضٍ  

“The Messenger of Allah ﷺ commanded us to greet our Imam with Salam, and to greet one another with Salam.” (Using translation from Ibn Mājah 922)   

أَمَرَنَا رَسُولُ اللهِ ﷺ أَنْ نُسَلِّمَ عَلَى أَئِمَّتِنَا وَأَنْ يُسَلِّمَ بَعْضُنَا عَلَى بَعْضٍ  

[Machine] After the Messenger of Allah ﷺ commanded us that when in the middle of the prayer or at the end of it, we should start before the tasleem by saying the greetings, praises, prayers, and dominion belong to Allah, then send peace on the right side, then send peace on your reciter and on yourselves. This is an indication that responding to the imam means that he intends to respond in his tasleem from the prayer, not that he starts it individually. It was narrated from Ibn 'Umar that he used to say, "Peace be upon you" on his right side, then he would respond to the imam. If someone greets him on his left side, he responds to him. And it was narrated from Az-Zuhri that he said, "Responding to the imam's peace is a Sunnah." And Yahya bin Sa'id al-Ansari used to say, "When you send peace on your right side, you are exempt from responding to him."  

بْنِ جُنْدُبٍ أَمَّا بَعْدَ أَمَرَنَا رَسُولُ اللهِ ﷺ إِذَا كَانَ فِي وَسَطِ الصَّلَاةِ أَوْ حِينَ انْقِضَائِهَا فَابْدَءُوا قَبْلَ التَّسْلِيمِ فَقُولُوا التَّحِيَّاتُ وَالطَّيِّبَاتُ وَالصَّلَوَاتُ وَالْمُلْكُ لِلَّهِ ثُمَّ سَلِّمُوا عَلَى الْيَمِينِ ثُمَّ سَلِّمُوا عَلَى قَارِئِكُمْ وَعَلَى أَنْفُسِكُمْ وَفِي هَذِهِ دَلَالَةٌ عَلَى أَنَّ الْمُرَادَ بِالرَّدِّ عَلَى الْإِمَامِ أَنْ يَنْوِيَ فِي تَسْلِيمِهِ عَنِ الصَّلَاةِ الرَّدَّ عَلَيْهِ لَا أَنَّهُ يُفْرِدُهُ وَرُوِّينَا عَنِ ابْنِ عُمَرَ ؓ أَنَّهُ كَانَ يَقُولُ السَّلَامُ عَلَيْكُمْ عَنْ يَمِينِهِ ثُمَّ يَرُدُّ عَلَى الْإِمَامِ فَإِنْ سَلَّمَ عَلَيْهِ أَحَدٌ عَنْ يَسَارِهِ رَدَّ عَلَيْهِ وَرُوِّينَا عَنِ الزُّهْرِيِّ أَنَّهُ قَالَ الرَّدُّ عَلَى الْإِمَامِ سَلَامُهُ سُنَّةٌ وَكَانَ يَحْيَى بْنُ سَعِيدٍ الْأَنْصَارِيُّ يَقُولُ إِذَا سَلَّمْتَ عَنْ يَمِينِكَ أَجْزَأَكَ مِنَ الرَّدِّ عَلَيْهِ