Muslim

Hadith Commentary:
(1)
Imam Bukhari rahimahullah holds the view that when a dhimmi (non-Muslim under Muslim protection) or a person under treaty does not openly revile or insult the Messenger of Allah sallallahu alayhi wa sallam, but instead vents his anger through hints and insinuations, then he should not be killed. This is as explicitly stated in this hadith, where the Messenger of Allah sallallahu alayhi wa sallam did not permit the killing of the aforementioned Jew.

(2)
Hafiz Ibn Hajar rahimahullah writes that the Messenger of Allah sallallahu alayhi wa sallam did not have the aforementioned Jew killed either due to the benefit of winning his heart (ta’lif al-qalb), or because he did not clearly and explicitly revile or insult. It is also possible that both reasons were present, and this is the better view. (Fath al-Bari: 12/352)

If a person openly abuses or reviles the Messenger of Allah sallallahu alayhi wa sallam, then there is consensus (ijma‘) on the obligation to kill such a person. As Ibn Abbas radi Allahu anhuma narrates, a slave woman (umm walad) of a blind Companion used to curse and insult the Messenger of Allah sallallahu alayhi wa sallam and would commit disrespect against him. One day, while she was insulting the Messenger of Allah sallallahu alayhi wa sallam in front of her master, he killed her.

When the Messenger of Allah sallallahu alayhi wa sallam came to know of this, he publicly declared:
“Beware! Bear witness that the blood of this slave woman is wasted and lost (i.e., there is no retaliation for it).”
(Sunan Abi Dawud, Hudud, Hadith: 4361)

When this ruling is established in the right of the Messenger of Allah sallallahu alayhi wa sallam, then it is even more necessary to kill the wretched person who insults Allah, His Book, Islam, the purified Sunnah, or the religion of Islam.

This is a reality that requires no further evidence.

Hadith Commentary:
This also pertains to a specific incident when a Jew explicitly used words of curse instead of the greeting of peace (salam). In today's era, if a non-Muslim greets with good words or a supplication, then one should also respond with good words. In the verse: ﴿وَإِذَا حُيِّيتُمْ بِتَحِيَّةٍ فَحَيُّوا بِأَحْسَنَ مِنْهَا أَوْ رُدُّوهَا﴾, there is a general command.

Hadith Commentary:
(1)
In some other hadiths, the response to the greeting (salam) of the disbelievers is mentioned as only "alaykum," that is, without the "wa" (and). Some scholars have offered subtle insights regarding responding to salam with or without the conjunctive "wa," however, both forms are correct and the narrations support this. At times, the Messenger of Allah (sallallahu alayhi wa sallam) also adopted this manner in response to the salam of the Companions (radi Allahu anhum), as when the one who performed his prayer incorrectly (musi’ al-salat) greeted him, he replied, "wa ‘alayk" (and upon you).
(Jami‘ al-Tirmidhi, al-Isti’dhan, Hadith: 2692) (2)
Now, this manner has become well-known for responding to the People of the Book, therefore, this style should not be adopted in response to a Muslim’s greeting. Thus, it is narrated from Anas (radi Allahu anhu): "We were commanded that in response to the greeting of the People of the Book, we should not add anything to the words 'wa ‘alaykum.'"
(Musnad Ahmad: 113/3)
Hafiz Ibn Hajar (rahimahullah) has also mentioned this matter.
(Fath al-Bari: 56/13)

Explanation:
1:
When these Jews come to you, they greet you in a manner in which Allah has not commanded you to greet (see al-Mujadila: 8).

Benefits and Issues:
In some other ahadith, the reply to the greeting (salam) of the disbelievers is mentioned as "alaykum" only, that is, without the "wa" (and).