Like the guy would set fire to a house, burn a child alive, and admit to it? He ran afoul of the 'imputed malice' legal concept, which states that if you plan and commit a crime, you are responsible for any side effects as if you had planned them as well. Say you hold up a bank, and run down and kill a person as you are getting away - you are guilty of premeditated murder. The theory being that if you hadn't committed the first crime, the rest would not have happened. Setting fire to a house is even more reprehensible - there is a reasonable expectation that someone might be inside.
I'm no expert in death penalty law, but generally speaking, it is applied in cases where the murder was planned, multiple deaths are involved, kidnapping and murder (indicating premeditation) is involved, or the murder is committed in a particularly horrific fashion - such as burning a child alive, murdering your pregnant wife, torture/rape and murder, things like that.
Some of this is determined by the state in which the crime was committed - not all states have the death penalty. Here is a hint - if you are going to commit murder in the US, don't do it in Texas or Florida. In premeditation cases, sometimes the person plea bargains their way to life, if the evidence isn't strong enough to assure a capital conviction. Hired killers often escape the needle by rolling over on the person who hired them - and that person usually gets the death penalty. The old man that Arnold wouldn't commute hired someone to commit multiple murders, though in that case the actual killer didn't roll over - he got the death penalty, too.
In the end, I'd have to agree that it doesn't really make much difference, one way or another, whether a nation has a death penalty or not. It is not a deterrent to crime. Probably more vengance than anything. I can't say I'm comfortable with letting the government decide life and death, but when I look at a few death penalty cases, and stifle the urge to puke when I read what those people did, I just don't feel motivated to political action.