“stone” = stoning was specified for the case of a betrothed woman guilty of adultery (Deut 22:23-24). In other cases just death was commanded without reference to its method (Lev 20:10; Deut 22:22). The Talmud calls for strangulation as the method of execution.

Deut 22:23-24  If a man happens to meet in a town a virgin pledged to be married and he sleeps with her, you shall take both of them to the gate of that town and stone them to death—the girl because she was in a town and did not scream for help, and the man because he violated another man’s wife. You must purge the evil from among you.

Lev 20:10  “‘If a man commits adultery with another man’s wife— with the wife of his neighbour—both the adulterer and the adulteress must be put to death.

Deut 22:22  If a man is found sleeping with another man’s wife, both the man who slept with her and the woman must die. You must purge the evil from Israel.

There is no indication of a trial taking place or even being planned. Although some measure of self-governing was allowed by the Romans, the death penalty could not be invoked by anyone but the Romans themselves, and this was not such a serious crime to the Romans that they would order death. The scribes and Pharisees are calling for a “lynching” (as Morris puts it) which may be the only method of securing capital punishment.