Or not, as the case may be. What kicks off the death of the arbiter? I have retrieved the horn and rescued Greene from the elves. I am not sure what I need to do to get the plot to progress.

Time. All of the quests that Catherine and Elfface give you are actually optional, the Arbiter's death will be triggered after a certain amount of time has passed. It should take a couple of months after the war starts.

Thanks.