Ah yes, you flew into the air to give a birds' eye "definition" (didn't really define anything) and never came back down to land on where, specifically, an injustice would occur in giving the death penalty to a murderer lol.
I just went and reread it. Is it because, as you said, we should "work on it" and seek rehabilitation?
I would somewhat agree and do not think every murderer needs, or should receive the death penalty.
However, I would argue equally that someone can simultaneously deserve a punishment, but it be better for them not to refuse it out of mercy and a desire for their betterment.
An example of this would be a father toward his son, who had struck his mother; the father may take mercy on the son and not ground him and instead only have him apologize because it would be better for his behavior.
Does the son still deserve punishment? Yes. Is it justified that the father, through his authority relieves the debter of his debt? Also yes.
Similarly, all murderers deserve to give their life in exchange for the life they took, but not all may need to do such in order to repent and improve.
If what you say is true, then what accounts for these laws and social structures exactly? Why not have no laws or any laws?
In truth, it is laws and "social structure" (whatever that is supposed to mean) that presume fundamental aspects inherit to human nature such as this right to life that every human has, which makes murder evil to begin with. Outside of these facts about human nature there exists no laws or social structure at all; the laws and the social structure assume these things to exist.