The tripod is truly the weakest political structure
The balance of power in a political tripod is only stable as long as everyone wants to win and all players would rather prefer the status quo to a high risk game.
In game theory this is the typical situation for board games with 3 players, where only one player will win the game. The two weaker players will always gang up on the strongest player so that player doesn't win.
But a real world political tripod is a lot different. Because there is no universal rules of "winning" there can be a lot of scenarios where two players gang up on the third to eliminate that player. As soon as two players become power hungry enough they can ally and destroy the third to split that player's goods.
Another common scenario is the strongest player convincing the weakest to destroy the runner-up. In this case the weakest player will become the second in command to the new emperor after the only real danger (the second strongest) is destroyed.
The third scenario which is likely to occur is when two players hide their true strength and each one is confident they can take the other in a 1-on-1 confrontation. Then they will gang up on the third player, eliminate that player, and hope to take power in the resulting face-off.
It is ironic that the political tripod is in contrast to the strong mechanical tripod actually the weakest political structure.
It takes only a single common cause for two players to destroy the third, because the forces of two combined are most likely enough to overpower the third without much risk.
A single absolute power is quite stable (the 1000 year reign of Leto). Two powers keeping each other in check is also quite stable, because each one would have to risk everything to attack the other one alone. One side needs a massive advantage to feel at least twice as strong as the enemy to win in a quick assault.
If you have more than three members you need a bigger number of allies to band together to eliminate a single member, as soon as there are three members in an alliance the alliance is internally a tripod and becomes unstable - each suspecting the others of treachery. Also it is harder to find a common cause to unify more than two factions without infighting. And without a vast majority it becomes too risky to try and take on 2 player together with 3 players.
The tripod is the only structure where a single alliance between two players is enough to almost guarantee victory over all other factions (namely the third player)