Sanhedrin 76b - The Mitzvah to Kill a Killer

The mishna lists murder among the capital punishments. In actuality they are specifically commanded to kill a murderer.

The Ramban in his additions to Sefer HaMitzvos (mitzvah 13) says that there is a specific mitzvah to not have compassion on one who murdered. The concept of the mitzvah is that Beis Din is obligated to carry out the judgement on the killer, not just as a positive mitzvah, but even as a lo ta'aseh as we find by a michasheifa.

The Brisker Rav (Hilchos Rotzei'ach) is troubled why the Ramban needs to count this as a separate mitzvah, since the Rambam (mitzvah 279) mentions this exact la'v of "lo ta'chos ei'necha a'lav" within the context of a judge not having compassion on one who committed a crime. The Brisker Rav proves from the Rambam in that he understands the nature of the issur very differently than the Ramban. In Hilchos Sanhedrin (14:3) the Rambam writes that a beis din who doesn't kill those who are chayev mi'sah are in violation of an a'seh of "u'bi'arta ha'ra" but not in violation of a lo ta'aseh. It is only by a mechasheif that we find a specific lo ta'aseh that the beis din would be in violation of if they fail to kill him. The pasuk of "lo ta'chos ei'necha" is mentioned by the Rambam in Hilchos Dayanim (20:4) in the context of their sentencing of a murderer. The beis din is not allowed to have compassion and say that a life has already been lost, so what use is there to take another life. Rather, they must overcome their misplaced compassion and sentence the murderer to death.

According to the Rambam the prohibition of "lo ta'chos ei'necha" is in the sentencing stage. That is why the Rambam places the halacha in hilchos dayanim. The violation is not by their refusal to kill him, rather by the compassion that they exhibit in the sentencing. This is also apparent from the Rambam incorporating other penalties under the same heading of this l'av, because the nature of the prohibition is not to have compassion in their sentencing. The Ramban, on the other hand considers this to be a l'av that compliments the a'seh of "u'biarta ha'ra" which applies after he is already sentenced. The Ramban holds that if the beis din fails to carry out the sentence, they are in violation of this l'av. The Ramban adds a separate mitzvah which is not counted by the Rambam, which is not to have compassion in carrying out capital punishment after it is decided.