Workplace disputes are never pretty, and it's easy to lose your cool under the pressure of mediating between warring factions. As the leader, it's your responsibility to keep the dialogue open and set the emotional temperature of the room. You have to lead by example, which means controlling your own feelings, talking people down if things get heated and never cracking under the stress of the situation.
Emotions can be difficult to control under pressure, so ensure you've developed a back-up plan. Breaking the session when things get too hot is a good tactic. This gives you and everyone involved time to take a breather, cool off and talk things over with the team. It acts as a mental reset break.
If you need to, a break is also an opportunity to call for reinforcements and bring in another senior figure to lend their support and help you make your final decision. This is, in effect, a negotiation process and you need to bring your negotiating skills to the table. You're trying to persuade two people with opposing views to find a middle ground, and you have to be level-headed to do that.
Finally, someone is always going to feel they came out of the dispute resolution having lost their argument. That can cause trouble which requires you to control your emotions both in the immediate and longer-term aftermath. They may be angry and they may set out to cause problems, one way or another. Always rise above any potential fallout and set the tone of future discussions by keeping your cool.
Set the emotional temperature of the room
Take a break when things get heated
Call for back-up if necessary
Deploy your negotiation skills
Rise above any fallout from your decision