In addition, why was the FBI, a domestic police agency, involved in an overseas terrorism case?
The FBI has a counterterrorist branch. My understanding is that part of the post-9/11 investigation revealed that the communication and investigation “wall” between CIA counterterrorism and FBI counterterrorism created a weakness ... particularly when overseas terrorists become a domestic threat.
My understanding is that the FBI is now more involved in overseas counterterrorism than they used to be.
I believe an embassy is US soil.