How would it be possible to prove that somebody is unrepentant? If a murderer shows remorse would that rule out the death penalty? How long should we wait to see if the murderer is remorseful - a year, ten years?
For me the Bible is not an authority, though I recognise that it is for many people. I do not know what the Bible has to say about the death penalty, if anything; as it is mentioned here I would be interested to learn.
Well, outside of the Scriptures themselves, I believe it's possible to talk to the murderer(I do not know prison laws). Should a psychological expert be able to speak with a proven murderer, they should be able to determine whether or not the person is remorseful for their crime or not. If they are, I believe the murderer's fate should be decided by those closest to the victim. Jail is, unfortunately, a prime place for criminals to become much worse than they previously were, and I would not recommend it unless the punishment were solitary confinement.
As for the Bible, I provide this example from Exodus 21. Not all the verses are provided, only those to do with the punishment of death for murder. But you can look it up yourself should you like to.
Exodus 21:12-14: "Whoever strikes someone so that he does must surely be put to death. But if he does not do it with premeditation, but it happens by accident, then I(the Lord) will appoint for you a place where he may flee. But if a man willfully attacks his neighbor to kill him cunningly, you will take him even from My altar that he may die."
Exodus 21:22-25: "If men fight and hit a pregnant woman and her child is born prematurely, but there is no serious injury, he will surely be punished in accordance with what the woman's husband demands of him, and he will pay what the court decides. But if there is serious injury, then you will give a life for a life, eye for eye, tooth for tooth, hand for hand, foot for foot, burn for burn, wound for wound, bruise for bruise."
There are numerous cases where a crime that is not murder is committed and death is still the punishment. I am not citing these verses because they don't match the context of those discussion.