Josephus reports in Antiquities that the High Priest Ananus was deposed in CE 62 by Herod Agrippa for having people stoned to death. This was in the temporary absence of a Roman procurator whose authority would normally have been required. The notice states that among the group was a man named Jacob (now rendered as James), ‘the brother of Jesus who was called the Christ’.
That would seem to settle it: the Sadducee elite killed James.
There are other sources indicating that James may have been hounded to death: pelted with stones and pushed down the Temple steps. But these do not imply a formal trial or execution.
The notice in Josephus therefore needs to be considered with caution. It is a report of a stoning, but it may well not refer to the same Jacob/James. Origen stated that Josephus did not believe Jesus was the Messiah and, indeed, that’s a fair conclusion from the general slant of Josephus’ writings. The phrase ‘who was called the Christ’ or even the whole description ‘the brother of Jesus who was called the Christ’ reads better as a marginal annotation by a later reader, who had drawn this conclusion, that was then incorporated into the text when it was recopied.
Jacob/James and Jesus were not uncommon names. A few lines later in the text, there is a probably coincidental notice of Jesus the son of Damnaeus who was appointed High Priest to replace the deposed Ananus.
Another reason for believing that the Jacob/James described here was not James the brother of Jesus is that the latter, by all accounts, was a popular figure with followers, according to Acts and the Pseudoclementines numbering in the thousands or tens of thousands. He had the authority to send letters giving instructions for gentile godfearers in the diaspora. Several early church authorities report him as acting as a people’s High Priest, in opposition to the Roman/Herodian appointees.
James died in CE 62, or possibly a few years later. Ananus was popularly elected as joint administrator of Jerusalem at the outbreak of the Jewish uprising in CE 66. It seems hardly likely that he would have been so chosen, had he been responsible for the death of James.
A more likely candidate would perhaps have been Ananias, son of Nebedeus, who was High Priest from around CE 48 – 58, and who (according to Josephus) took bribes, misappropriated tithes due to lower order priests and was generally greedy and violent. It was Ananias who robustly interrogated Paul/Saul. Ananias continued to have influence after his tenure as High Priest ended (and more of him, later).
A similar name to Ananus; conceivably part of the reason an early commentator, who might have had other sources to draw on, jumped to the conclusion that the Jacob/James, described as having been stoned to death in Antiquities, was James the brother of Jesus. And then wrote this in the margin of his or her copy.
Josephus, former zealot general captured and employed by the Romans to write a history of the Jewish War, demonstrating the futility of resistance to other subject peoples, was in a delicate position. Did he to an extent cover his tracks and those of a messianic movement that incorporated a submerged Davidic dynasty? (See ch 10 of Jesus the terrorist)
In which case, the notice about the stoning of possibly another Jacob/James in Antiquities, is really a distraction. Josephus cut out or kept out the details of how James died.