RIVERSIDE, Calif. — Both the prosecution and the defense involved in a trial set to start here on Monday basically agree on the following: Before dawn on May 1, 2011, 10-year-old Joseph Hall went to his family’s living room armed with a snub-nosed revolver, pointed it at his father’s head as he lay sleeping on the couch, and shot and killed him.

From there, the two sides are likely to differ on both the events that preceded the shooting and Joseph’s exact motive, elements complicated by his age and the fact that his father, Jeff Hall, was a rabid neo-Nazi. And those facts raise several more philosophical quandaries that, depending on how the judge weighs the answers, may determine the outcome of the trial. Among them: whether virulent racism can amount to parental abuse, whether a child exposed to such hate can understand the difference between right and wrong, and whether someone who grows up in such toxic circumstances can be blamed for wanting a way out.

The prosecutor, Michael Soccio, says that the actions of Joseph Hall have little to do with Nazism, but rather with his anger at being punished and spanked by his father at a party the day before the killing and the boy’s worries that his father would leave his family. Though he says he sympathizes with Joseph and his upbringing — “There’s a sweet side to him,” Mr. Soccio said in an interview this month — he also has little doubt that the boy is a killer.

“What he did, had it been done by anybody older, there would be no doubt that it was a murder,” said Mr. Soccio, the chief deputy district attorney in Riverside County. “It’s planned. It’s premeditated. It was carried out in a cold, killing fashion. It is a murder.”