The Arch Magi were the branch of House Illistim which was directly responsible for serving as the royal bodyguards. Though their primary duty was to protect the Argent Mirror, they would also have other charges, such as the Council of Thrones or the royal family. They have even been dispatched to serve alongside the Sapphire Guard at times. Consequently, they were required to answer to multiple sources, not serving solely the Queen like the Elemancers.
Arch Mages were made for the single purpose of pure destruction, with all of their training and focus directed toward that end. Their research, such that it existed, was based on more effectively blowing things up. They were considered elite military units for this reason, and had nothing resembling the independence of the Elemancers.
While primarily serving the role of bodyguards, given assassination attempts, they were assigned sometimes to serve as defense consultants. These would typically be posts with the Sapphire Guard, whether to serve as extreme tactical advantage, or support the wardings that guarded Ta'Illistim. Being solely focused on magical combat and defense, they would be a major buttress to the magical units within the Sapphire Guard. They were also known to consult with the Artificers, most obviously for their command of the airship fleet, which was known to be used for military purposes.
"Illistim Culture - Part II: Branches of the State" - Haethera Illuvassas Illistim (5109 Modern Era)