that he asked Anas bin Malik: "Did the Companions of the Messenger of Allah (ﷺ) use to shake each other's hand?" He said, "Yes."
Jami` at-Tirmidhi Book 42, Hadith 42
← Previous Hadith that a man said: "O Messenger of Allah! When a man among us meets his brother or his friend should he bow to him?" He said: "No." The man continued: "Should he embrace him and kiss him?" He said: "No." He said: "Should he take his hand and shake it?" He said: "Yes."

Jami` at-Tirmidhi Book 42, Hadith 41 Next Hadith → that the Prophet (ﷺ) said: "taking hold of the hand is from the completeness of the greeting."

Jami` at-Tirmidhi Book 42, Hadith 43