Tetrarch
Tetrarch is the title held by each of the four rulers of the Remunan Kingdom
Requirements
The only requirement to be a Tetrarch is to be the oldest son or daughter of the one of the previous Tetrarchs.
Appointment
New Tetrarchs begin their reign when one of their predecessors dies. Then the remaining 3 Tetrarchs are obligated to either abdicate or commit suicide. Death of a Tetrarch is considered a sign that a new generation should take over.
"Father, what are you doing?"
"Tetrarch Antonius has died. It's time for the new generation to take over"
"But, father, you don't need to die. You can abdicate"
"No Tetrarch from our family had ever abdicated. We believe that a Tetrarch can be truly independant only if his direct predecessor dies. Remember that, son."
If a Tetrarch doesn't have children the title goes to one of his siblings or a member of the extended family. However, it must be someone who is related by blood to the Tetrarch.
Marriage Rules
Marriages between Tetrarchs or members of their houses are strictly forbidden. It is a way of avoiding all four Tetrarchs being from the same family. If someone attempts to break that law, the nobility has the right to rebel, depose the Tetrarchs and execute the couple which attempted to get married.
Tetrarchs and their families can marry people from the remunan nobility or members of foreign noble and royal houses.
Really interesting that when one dies, all four are replaced. I like the bit of prose you've put there illustrating that, though abdication is an option, some choose suicide instead.