Progression of changes in Salat
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Something interesting I noticed was in Al-Muwatta of Malik ibn Anas it says
"Book 3, Number 3.14.57
Yahya related to me from Malik from Nafi that Abdullah ibn Umar used to say the tashahhud saying, "In the name of Allah. Greetings belong to Allah. Prayers belong to Allah. Pure actions belong to Allah. Peace be on the Prophet and the mercy of Allah and His blessings. Peace be on us and on the slaves of Allah who are salihun. I testify that there is no god except Allah. I testify that Muhammad is the Messenger of Allah."
"Bismillah, at-tahiyatu lillah, as-salawatu lillah, az-zakiyatu lillah. As-salamu ala'n-nabiyyi wa rahmatullahi wa barakatuhu. As-salamu alayna wa ala ibadi'llahi's-salihin. Shahidtu an la ilaha illallah. Shahidtu anna Muhammadu'r-rasulu'llah."
He used to say this after the first two rakas and he would make supplication with whatever seemed fit to him when the tashahhud was completed. When he sat at the end of the prayer, he did the tashahhud in a similar manner, except that after the tashahhud he made supplication with whatever seemed fit to him. When he had completed the tashahhud and intended to say the taslim, he said, "Peace be on the Prophet and His mercy and blessings. Peace be upon us and on the slaves of Allah who are salihun."
"As-salamu ala'n-nabiyyi wa rahmatu'llahi wa barakatuhu. As-salamu alayna wa ala ibadi'llahi'ssalihin ."
He then said, "Peace be upon you" to his right, and would return the greeting to the imam, and if anyone said "Peace be upon you" from his left he would return the greeting to him."
So in the Al-Muwatta (the earliest collection) you are giving Salam towards your right and then to the Imam, and if the person to your left said "salam" you would return it.
Fast forward to later traditions and the Salam is to both the left and the right directed towards the Angels that record your good and bad deeds.
It is interesting to see how ideas progressed and were added on to.