The Messenger of Allah ﷺ used to say the takbir with every lowering and rising, and he would give the salam to his right and to his left. Abu Bakr and Umar used to do the same.
أخبرنا عمرو بن علي قال حدثنا معاذ ويحيى قالا حدثنا زهير قال حدثني أبو إسحق عن عبد الرحمن بن الأسود عن علقمة والأسود
كان رسول الله ﷺ يكبر في كل خفض ورفع ويسلم عن يمينه وعن يساره وكان أبو بكر وعمر رضي الله عنهما يفعلانه
[Sahih] Ahmad narrated it 1:386 from Yahya Qattan through this chain, and it is also in Al-Kubra, no. 670. Tirmidhi mentioned it in the chapter on prayer, in the section on what has been said about the takbir at ruku and sujud, no. 253, and said, "Hasan sahih." The hadith has very many corroborating narrations.