Tafseer Ibn Kathir . The Noble Qur’an An-Nisa 4:86
(When you are greeted with a greeting, greet in return with what is better than it, or (at least) return it equally.) meaning, if the Muslim greets you with the Salam, then return the greeting with a better Salam, or at least equal to the Salam that was given. Therefore, the better Salam is recommended, while returning it equally is an obligation.
Imam Ahmad recorded that Abu Raja’ Al-`Utaridi said that `Imran bin Husayn said that a man came to the Messenger of Allah and said, “As-Salamu `Alaykum”. The Prophet returned the greeting, and after the man sat down he said, “Ten.” Another man came and said, “As-Salamu `Alaykum wa Rahmatullah, O Allah’s Messenger.” The Prophet returned the greeting, and after the man sat down he said, “Twenty.” Then another man came and said, “As-Salamu `Alaykum wa Rahmatullah wa Barakatuh.” The Prophet returned the greeting, and after the man sat down he said, “Thirty.” This is the narration recorded by Abu Dawud. At-Tirmidhi, An-Nasa’i and Al-Bazzar also recorded it. At-Tirmidhi said, “Hasan Gharib”.
There are several other Hadiths on this subject from Abu Sa`id, `Ali, and Sahl bin Hanif. When the Muslim is greeted with the full form of Salam, he is obliged to return the greeting equally.
As for Ahl Adh-Dhimmah the Salam should not be initiated nor should the greeting be added to when returning their greeting. Rather, as recorded in the Two Sahihs their greeting is returned to them equally.
Ibn `Umar narrated that the Messenger of Allah said,
(When the Jews greet you, one of them would say, `As-Samu `Alayka (death be unto you).’ Therefore, say, `Wa `Alayka (and the same to you).’) In his Sahih, Muslim recorded that Abu Hurayrah said that the Messenger of Allah said,
(Do not initiate greeting the Jews and Christians with the Salam, and when you pass by them on a road, force them to its narrowest path.) Abu Dawud recorded that Abu Hurayrah said that the Messenger of Allah said,
(By He in Whose Hand is my soul! You will not enter Paradise until you believe, and you will not believe until you love each other. Should I direct you to an action that would direct you to love each other Spread the Salam among yourselves.)