Swords of Sanghelios

Summary
The Covenant separatists are made up of the Elite (Sangheili), Grunt (Unggoy), Hunter (Lekgolo) and maybe some of the Jackal (Kig-Yar) castes within the Covenant's hierarchical society. This faction was created at the begining of the Civil War of the Covenant and since then has been fighting the remainder of the Covenant, led by the Prophets and their Brute, Drone, most and maybe all Jackal and loyal Grunt supporters. It is speculated that this disgruntled faction is lead by Half-Jaw and/or the Arbiter. Fronts of this war include High Charity, the Covenant's slipspace-capable capital, Delta Halo, and areas of space where Covenant ships operate.

(This is not necessarily the first internal dispute in the Covenant's society; when the current Arbiter was assigned to his office, he was reminded by the High Prophet of Mercy that an Arbiter was necessary during the Grunt Rebellion, which possibly took place in the Age of Doubt and was a conflict between the Covenant and its lowest caste, the Grunts or Unggoy.)

How the faction comes to life
The High Prophet of Regret led a botched mission to Earth and was forced into The First Battle of Earth upon discovering what turns out to be the UNSC homeworld. He escapes to Delta Halo, but is killed by the Master Chief, which is considered a significant failure in the Elites' mission to protect the leading caste of the Covenant. The Master Chief was assigned on a mission to capture and later kill a member of that caste. The remaining Prophets, led by the High Prophet of Truth, use this opportunity to replace the Elites with Brutes as their guards. However, this move effectively shames and angers the Elites, who have grown accustomed to their comfortable position in the caste system, and they threaten to resign from the High Council.

Later, the Elite councillors are assassinated by the Brutes at the behest of the Prophets: this purge, along with the understanding of what the Halos actually are meant to do, help divide the Covenant along the lines seen in the war. Instead of a Great Journey, the Gravemind's revelation regarding the superweapon contradicts the Covenant's belief system and is motivating at least the Arbiter in his eventual opposition to the Convenant as they try to activate Delta Halo later in the game.

Military Forces: The Elites and their supporters
Commanded by Half-Jaw and the Arbiter. Includes Elite, Grunt and Hunter castes, along with loyal ships of various classes to their cause. Also involved in this faction is the UNSC's most effective Spartan, the Master Chief; through an alliance of convenience, he is meant to stop the Prophet of Truth and his guards at the behest of the Gravemind, the effective leader of the Flood. The Arbiter, meanwhile, is meant to recover the Index and prevent its activation by the Prophets and their proxies.