You do not have to be the mediator, nor take a side when your partners are in conflict with one another. Just because you are in a triad, quad, or any other polyam dynamic does not mean you have to engage in every argument.
You've Decided Not to Engage
If your partners are in the middle of a conflict, it can place a damper on planned events, experiences, and the relationship as a whole, depending on the severity and length of the argument. For those reasons, you may feel inclined to mediate between them or attempt to diffuse the situation; however, this may not be the best choice. There is a chance that instead of your partners learning conflict resolution skills, they become codependent on your mediation. You may also become the person used to sway the argument in one partner's favor which can create resentment. Maybe your input is not valued and feels like you are butting in. If you step into conflicts "laying down the law" (telling everyone to stop completely or what will be done in the situation), your partners may feel like they are being bullied or that their voices aren't being heard.
Express How You Feel
Your partners may not be fully aware of how you feel when they argue and pull you into it. Let them know that you feel like you have to pick between them, even if one is correct. Make them aware of how long it takes during and after an argument for them to cool down and how that affects the flow of the relationship and planned activities.
Setting the Tone
There is no wrong time to set the tone for how to interact with your partners during an argument they are having separate from you. Let them know as a part of your boundaries that you would instead not be included or affected by disagreements in the polycule (triad, quad, etc.) unless necessary and with your consent. If they have a habit of fighting - suggest they seek out 3rd party mediation. Be clear about what that means to you.
No passive-aggressive, derisive, or derogatory comments about the other partner in your presence while in the middle of a conflict
Respecting that even if you have an opinion, you have the right to reserve it, so you do not appear to be taking sides
You can seek a safe space - another room/home, time alone, focusing on other activities while your partners resolve their conflict to protect your peace
Do not include yourself in any passive-aggressive behaviors, including "punishments."
Respecting that you may not feel comfortable being intimate until everything is settled
Stick to Your Plan
You have to be responsible for ensuring you are also not making the situation worse when your partners are fighting. Every fight's escalation may not be your fault. Still, you could be the reason for miscommunication, someone feeling like they are being ganged up on, and a two-way disagreement leaking over into the entire polycule.
Do not deliver messages between the parties - they can text each other or call if they are too heated to speak face to face
If you choose to listen to your parties vent about the situation, only respond to their feelings
do not speak negatively about your other partner - say, "I understand you are hurt," or "You're angry, and I can see why."
Avoid gossiping about the disagreement with friends and partners - this may get back to your partners, and it may enlist the unneeded intervention of others
When to Intervene
If you observe or become aware of acts of domestic violence in the middle of an argument
verbal, emotional, financial, and physical abuse
The structure of the home (if nested) or polycule is falling apart because your partners refuse to conclude a disagreement.
The affected parties are unaware of the information that would solve their problem.
Your partners need the mediation - you can mediate IF you are trusted to be neutral
this is not a common reoccurrence