The Triumvirate is a political regime dominated by the powerful individuals. It is, in effect, a three-man directorate with dictatorial powers that share in the governing of an empire. Each are meant to be equal in status and authority, and meant to keep the other person in check.
In reality, this is often not the case. The most famous of these alliances was the Triumvirate between Caesar, Pompey, and Crassus. Each of them were treacherous and egomaniacal, and the agreement ended after Crassus got himself killed. Caesar and Pompey turned on each other which ended up in a civil war for control of Rome. The second was between Augustus, Mark Antony, and Lepidus. This also ended up in civil war which led to the death of the Roman republic.
The pattern being set is that the rule of three tends to end in backstabbing treachery, as one individual always wants more power. There needs to be some mechanism that re-balances power so that no two members of the group can become more powerful tha the last one. Is there a way that this political system could be made to function over the long term without devolving into backstabbing treachery?