Who runs the Jedi Council

Does the Jedi Council have a higher military authority than the Supreme Chancellor?

My existing preliminary answer was deleted for some reason, but based on the feedback and some other information here on SE, I have formulated a better one anyway.

First of all, the best-chosen answer to this question shows that the Jedi are essentially advisors when it comes to official Republic structures. You can take on roles in the army, but ultimately they are under the authority of the council, not the republic per se. The other answer also suggests that the council is an authority on its own so I think we can accept that.

This question and its answers also suggest that Jedi were appointed as generals outside of normal procedures and were of great value to the Republic, which would explain why the irregularities associated with this situation would be tolerated by the Republic. Of course, this was also an element in Palpatine's plan to disperse the Jedi so that they would be easier to destroy.

Given these two points, my informed, but not directly canonical, guess would be that since the Jedi have their own power structure (the With the republic cooperates, but not this one subject ), can be asked to serve as generals in the GAR, but are not required to accept such commissions. In Anakin's case, the council does not oppose the chancellor's order because the council has direct authority over him but the republic does not (i.e., he is actually a civilian when it comes to the authority of the republican military over him). You're not even officially involved. Anakin was asked to direct the operation; The council orders Anakin to decline the Chancellor's request, which he can do as a Jedi. Whether this rejection is its own decision or that of the council is irrelevant to the government of the republic.