Abu Dawud

Benefits and Issues:
➊ In other hadiths, it is mentioned that the Messenger of Allah (sallallahu alayhi wa sallam) said: "Look at how jealous Sa'd is, and I am more jealous than him, and Allah is the most jealous of all."
(Sahih al-Bukhari, Kitab al-Nikah, before hadith: 5220; Sahih Muslim: Kitab al-Li'an, hadith: 1498)
➋ In this hadith, it is stated that a believer should abide by the limits set by Allah and not transgress them.
In Islam, human life is given great value and importance, and even in such incidents, no one is permitted to commit murder. Rather, if there are four witnesses or a confession, then the punishment of stoning (rajm) will be carried out.
If there are no witnesses, nor a confession from the woman, but only the husband's claim, then in this case, stoning (rajm) will not be carried out; rather, li'an (mutual cursing) will take place.

Benefits and Issues:

➊
A person who witnesses a man or woman committing adultery in the very act does not have the right to kill them.

➋
In such a case, he should bring along the testimony of three other men, so that all four of them witness the act together.

➌
Upon completion of the testimony, the court will administer the prescribed Islamic punishment (the punishment of one hundred lashes) to such a man or woman.

➍
The wisdom in prescribing this standard of testimony is to prevent a situation where someone, out of personal anger, kills another and then claims afterward that he saw him/her committing adultery.

➎
If a man sees his own wife involved (in adultery), the path of divorce and li'an (mutual cursing) is available to him; therefore, taking the law into his own hands and killing his wife is not permissible.

➏
The statement of Sa'd bin 'Ubada radi Allahu anhu is an expression of his protective jealousy (ghayrah); thus, the Messenger of Allah sallallahu alayhi wa sallam praised his sense of honor, but did not grant him the authority to kill the offender himself.