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The way to start is to ask why people back stab. The answer is, mostly, to gain power and status.

The problem with any kind of monarchical government (regardless whether it's an old-fashioned hereditary kingship or a dictatorship or whatever) is that whoever is on top stays on top, and everyone else is secondary and stays secondary. The only way to power is to knife the people above you in the back, and the only way to retain power is to knife the ambitious people below you before they knife you.

The huge advantage of governments that have a regular, planned turnover of leaders is that ambitious people have a path to power that doesn't involve the same amount of backstabbing. It a huge advantage of democratic forms of government, but many oligarchic governments also share this feature. Note that this doesn't rely on would-be leaders in these sorts of governments being nicer than in autocracies -- it just requires that they see playing the political game as giving them a better chance at power than carving a bloody trail to the top would.

As long as people are people, there will be ruthless people seeking power. As long as your government doesn't have a succession plan which gives ruthless people a peaceful path to power, they will follow a non-peaceful path. And it doesn't sound like there's a peaceful path to permanent ultimate power for anyone in your state -- including the triumvirs themselves.

When you say "triumvirate" what you specifically mean is "three individuals." But triumvirates exist all over the place. For example, the U.S. triumvirate of the executive, legislative, and judicial branches. Similar triumvirates exist in most democracies or republics.

The problem is when you start talking about three individuals. The "system" or government is not an intelligent thing. It has no intrinsic power or authority. A code of laws can be drawn up that theoretically constrains three individuals holding equal but in aggregate ultimate power — but that's really meaningless since the power and authority is held by the individuals — not the laws.

Winston Churchill said, "No one pretends that democracy is perfect or all-wise. Indeed, it has been said that democracy is the worst form of Government except all those other forms that have been tried from time to time." Democracy is considered the worst form of government because government by consensus (or "the crowd") is unwieldy as too many people must be convinced to get anything done.

But it's better than the rest because convincing an individual is easy compared to convincing a crowd. And one strong-willed personality can corral two weaker-willed personalities no matter how equal their governmental authority or power is. ("Does the NRA have pictures of you golfing with Satan?" quips one of my favorite movies...)

So, to invoke Newton's Third Law, the only way to force a triumvirate of individuals to behave is to create an external force that polices their behavior and forces them to behave. Of course, you wouldn't have a triumvirate anymore.

Which is a long and fancy way of saying it can't be done with individuals. The closest you can get is to involve groups of people such that at least one of the three legs of power is painfully difficult to bring under individual control. AKA, democracy.

One simple solution is to include the concept of mob rule, IE the people acquiesce to be governed by a Triumvirate, however should the Triumvirate fail (either in a way written into whatever passes for a constitution in your country or simply in the eyes of the people) then the Triumvirate can and will be violently deposed (and probably beheaded) and a new Triumvirate installed. A simple rule for the failure of a Triumvirate might be: “If any one of the Triumvirs accuses the others of attempting to supplant them, all three must be removed from power”.

If the concept of Triumviracy is upheld but individual Triumvirates are subject to being replaced by a ‘democratic’ (One mob, one vote) process, then it is in the best interests of individual Triumvirs to make the system work. If you consider political systems like the rule of the British Monarchy (Technically a Monarchy, held together by a series of Gentleman’s Agreements that the People will do whatever the Monarch asks as long as the Monarch doesn’t ask them to anything) then this seems almost sane.

Now: this doesn’t exclude the possibility of other ambitious types actively trying to get rid of the current ruling triumvirate, but it does give your three rulers powerful incentive to keep each other alive and functioning.

Triumvirate as an individuals' agreement is highly unstable. Triumvirate as a political system is realistic.

As @JBH correctly laid out, a simple agreement to share power between three individuals wont work. What may work, however, is a political system that gives each of the three individuals a power which other two can't steal.

US is the most prominent example of three individual branches of government working together (or, sometimes, against each other). If we give more power to the Speaker of the House and to Chief Justice, we can see it as a triumvirate.

In a historical setting, we can imagine three bases of power, say, nobility, military and clergy, each represented by a leader, and three leaders forming a triumvirate. There were no real precedents for that which lasted as a long term setting, but I think it's possible if a country get used to it.

It sounds to me like what you’re talking about is a legal division of power between technical equals. The two big examples of this in the Roman Empire were the Tetrarchy and the later empire, when different emperors ruled the Eastern and Western halves of the empire. Unlike the triumvirates, there was a lot more stability between these groups, and they existed as separate entities for decades. Within a world-building context, that suggests to me that such power-sharing arrangements can last at the scale of decades. I’m not sure there’s a good example of this lasting for centuries, but I could look for some if that would help!

I’m not sure that the triumvirates are the best example of long-term, stable power-sharing: the the First Triumvirate was an alliance between powerful men, each of whom had a legal position from which they derived their powers (Caesar was senator, consul and then proconsul, Pompey and Crassus were senators, prefects and then consuls) but primarily it was an agreement to work "behind the scenes" to achieve their mutual aims. Not surprisingly, once those aims diverged, the triumvirate fell apart. So I think this comes under the general category of competing power centers within a single executive (e.g. Honorius and Stilicho).

The Second Triumvirate was a legal entity, but it was more similar to the Tetrarchy or the division of the Roman Empire: initially, it was pretty clear that Antony was dominant over the other two members, but eventually they decided to maintain different spheres of influence with Octavian taking the west and Antony taking the east. It only lasted for a single decade, however, and so probably isn't the best example for your needs.

The leader is only the leader because other people obeys their orders.

In actual democracies, there are laws even the leaders must follow. Any leaders who break these rules will no longer be obeyed as leaders.

Even in non-democracies there will be laws, maybe unwritten ones, that leaders must follow. There are limits to power. These laws are enforced by the people close to the leaders, regional leaders, chief bureaucrats, head priests etc. This goes all the way down to the common people who will revolt if things get too bad.

So, to get stability you need a strong feeling in the rest of the nation that there SHOULD and MUST be a triumvirate leading them. Any leader who tries go solo will face strong opposition from everybody.

It will be tricky to figure out the details of what these (possibly unwritten) laws should be. Drawing the line between normal power use and power abuse is hard.

I suggest that there are limits to what a single triumvir can do. In short, they should only give orders that are either reversible (like an arrest) or slow (like starting a new building). The point is that the others can countermand these orders if needed.

For irreversible orders, most notably executions, the whole triumvirate must agree.

Things will be more stable if each triumvir has their own power base.

The simplest is if there are three powerful families/clans that appoint one each. Others have suggested military/religion/merchants.

It is important that if a triumvirate dies, their power base will not be weakened. Instead people must think that "This is probably done by one of the others. How rude! Let us support the newly elected heir to show our disgust in this sort of behavior!"

Anyway, nothing in life is stable. If you claim that this system have been working for thousands of year, readers are going to have strong disbelief problems. Don't go overboard on the duration of your nation.

Anything can remain stable as long as it has sufficient legitimacy.

Legitimacy comes from buy-in by groups with power. That usually takes the form of the people deciding that the current government is the correct one (for whatever reason) or from having a large military presence that decides that the current government is the correct one, and then is prepared to enforce that on the people by force of arms. It can also come from having those who might benefit in the short term from replacing the government (like rivals to the ruling body, or the members of the ruling body itself) deciding that the governmental form has value in and of itself and should be respected.

Various ways to do this...

• Religious beliefs. The obvious one for this would be a sort of setup where the culture has three gods who rule in concert, and the members of the triumvirate each represent and/or embody one of those gods in some way.




• Tradition. Once you've managed to keep the system going for a generation or two, it's a source of societal stability. People like societal stability.




• Tremendously powerful individuals with long lives. Perhaps your triumvirate is composed of immortal vampires, who can easily overpower everyone else in the country and who happen to like each other. Alternately, it might be an outside power. Some larger country, a local dragon, the gods themselves... whatever. If there's someone who has the power to ensure that it keeps happening, and the persistent will to do it, you're set.




• The magic system. If your country is partially founded on access to a powerful ritual, and that ritual must have three focal individuals, who are in turn empowered by it, that'll tend to nudge them in the direction of persistent power-sharing. If having those individuals is vital to the wellbeing of the nation, and the ritual cannot be performed frequently, or requires unusual skills to either execute or take advantage of, then that cuts down on the churn of people trying to get into the position.

The problem with the Roman triumvirate was that each person did not have a constitutionally enforced, distinct and different source of authority. When you take Caesar, Pompey, and Crassus for example, they started out with spheres of influence that was just coincidental. Caesar had the support of the peasantry, Pompey had the support of the military, and Crassus had the support of the nobility. The problem was that any one of them could at any point make another one irrelevant if they also gained the support of one of those factions. Just by dying, Crassus's power was left to be dissolved into Caesar's and Pompey's spheres of influence. Then when Caesar conquered Gaul he threatened Pompey's favor with the military.

Imagine instead if each man had to be elected by a respective portion of a nation's population. If Rome decided that they would have elections where one triumvir was decided by each of these 3 distinct groups, then the power would not pass if one was ousted; so then, not only do you take away the motivation to oust a your fellow triumvirs, but doing so would anger that portion of the population whom elected and favored that person likely resulting in the election of a new triumvir that would be more anti-you for having tried to do so.

So, the result in ancient Rome would have been that when Crassius died, a new triumvir favored by the lords would have taken his seat, and when Caesar conquered Gaul, it would have given him the opportunity to run for future elections for triumvir of the military, but he could not hold the seat of triumvir of military and triumvir of the commoners at the same time; so, either way he'd be forced to share his power with someone else favored among one of these groups.

If Rome's original triumvirate had been a legal system of governance, it would have likely become a much more stable one. The biggest risk to this system's longevity is the risk that two parties may become irrevocably aligned against the 3rd. So, when deciding where the base of each triumvir's power comes from it is even more important than in other systems to divide the power bases on a division of imperatives and not divisions of geography or culture.

Examples:

• If you divide a country by regions like if you were to split the USA into the Northern (Old Union) states, Southern (Old Confederate) states, and Western states then you could create 3 distinctly different cultural areas that would have worked well to balance different viewpoints a few decades ago. However, as cultural shift has made the Western and Northern states more like each other than Southern states this would undermine the Southern triumvir's ability to represent their demographic. This also leads to the risk of each triumvir being seen as the ruler of his region which could lead to the country breaking apart into 3 separate states turning your nation into more of a confederation of dictatorships.


• If you divide a country by institutions you can avoid schisms a bit better, but have a similar problem with cultural shift. So if you have branches of Clergy, Military, and Commerce then they may be distinctly opposed in one generation, but as cultural values shift, you may see convergence where perhaps the Commerce and Clergy sector decide that Military is an immoral waste of funds and basically dissolve the third person's ability to represent their demographic.


• Dividing a Nation by imperative means each group has a distinctly different set of needs for well-being (Like Lower, Middle, and Upper Class). No matter how much culture shifts, the lower class will care most about issues of basic survival: welfare, minimum wage, etc. The middle class has survival covered, so they will care most about quality of life issues: tax rates, workers rights, etc. The upper class has quality of life covered; so, they will care most about things that could stimulate or undermine their businesses: trade deals, employer's rights, etc. Because the values are tied to states of being, cultural shift is unlikely to ever cause a major convergence. This means that a 3-way adversarial system is maintained so that no pair of them is ever going to benefit by always agreeing with another one.

I think that your question cannot be answered because you are starting from an ungrounded assumption. There is no stable form of government; it's human nature. There will always be someone or a faction or an elite vying for power, trying to prevail over the others. Even democracy is not stable, no matter how complex and sophisticated the system of check and balances is, it will always be on the verge of being toppled. The real democracy is the one that requires a perennial fight to be preserved. If you see no fight than it's the moment to wonder why.

If I understand your question correctly than the answer is law. Non-permanent political or social structures can be perpetuated by being enshrined in laws. Once a social or political position has been transformed into an institution through legislation, an individual can be killed but his or her power is then passed on to someone new.

and the agreement ended after Crassus got himself killed.

Indeed, the way it will remain stable is for them to all be equally powerful characters, equally popular and not inclined to any of the others (resulting in side-taking).

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.

Not if we're looking at a Roman-style setting. Whilst you could in theory keep the Senate/Government branch equal (subject to not doing stupid things a la Mark Antony), there is no way to force the legions/military to stick to your rules. And the name of the game in that context is often who controls the legions.