The Sunni tradition is based on the following ahadith:
(22)Chapter: The salam to exit the prayer when one has finished and how it is done
'Amir b. Sa'd reported:
I saw the Messenger of Allah (may peace be open him) pronouncing taslim on his right and on his left till I saw the whiteness of his cheek.
(83)Chapter: The Takbir when sitting up from prostration
It was narrated from 'Abdur-Rahman bin Al-Aswad from his father-and Alqamah-that:
Abdullah said: "I saw the Messenger of Allah (ﷺ) say the Takbir every time he went down and got up, or stood or sat; he said the Salam on his right and his left: 'As-salamu alaykum wa rahmatulah (peace be upon you and the mercy of Allah ),' until the whiteness of his cheek could be seen." He said: "And I saw Abu Bakr and 'Umar, may Allah (SWT) be pleased with them both, doing the same."
(340)Chapter: Regarding The Salam
Narrated Abdullah ibn Mas'ud:
The Prophet (ﷺ) used to give the salutation to his left and right sides until the whiteness of his cheek was seen, (saying: "Peace be upon you, and mercy of Allah" twice.
AbuDawud said: This is a version of the tradition reported by AbuSufyan. The version of Isra'il did not explain it. AbuDawud said: This tradition has been narrated by Zubayr from AbuIshaq and Yahya ibn Adam from Isra'il from AbuIshaq from AbdurRahman ibn al-Aswad from his father from Alqamah on the authority of Abdullah ibn Mas'ud. AbuDawud said: Shu'bah used to reject this tradition, the tradition narrated by AbuIshaq as coming from the Prophet (ﷺ).
And many more similar ones.