Ultimatum.
Vertex did not know where his sister developed such courage to talk back to him, and that too for a mere servant. Azula had never raised her voice at anyone and was always the most compassionate one among the three siblings.
Ignis was a goof most of the time, but he had very high goodwill among the commoners. He was the beloved first prince of the empire, especially in the capital, his words were like sacred commandments.
When he thought of himself, Vertex was only good at maintaining order. He would follow protocols for everything. He was seen as the most righteous person in the empire. While the first prince was beloved, the second prince was the most respected.
This sort of competition was the reason for this internal argument in the royal family. After Azula laid down her words, she was ready to go all out to face her elder brother. Fragrantia saw this and said, "Have I spent too much time away that my children have forgotten the rules in the dining hall?"