Although it will make people mind that their subordinates communicate with their leaders beyond their authority, after all, such an approach is to ignore their own behavior and report themselves in the dark. However, since the other party has already made such behavior, don't continue to pester this matter, otherwise it will make the other party get along very embarrassed and make the other party more dissatisfied with themselves.
I don't think it's necessary to deal with this employee deliberately. After all, it was his idea that the other party exceeded his authority. Although there are some shortcomings, talking more will only appear petty. As his boss, he can keep an eye on his movements at any time, have frequent heart-to-heart conversations with the other party, and eliminate the doubts of the other party, thus establishing deep feelings and trust. As the leader of the other party, there must be some reasons why the other party can do such behavior. He can self-examine whether there is anything wrong, and try to let the other party have something to say to him, instead of communicating beyond his authority, which will make him look incompetent.
The height of the leadership station is different, and the way and angle of thinking are different. Although I'm a little worried about it, I won't take it personally. I believe that time will prove everything, and the other party will gradually accept themselves, so such behavior will not happen. Don't punish employees when things happen, treat employees as your friends, so as to establish different relationships and get along well.
In the face of the ultra vires behavior of your employees, don't be too impulsive when communicating with leaders, and you won't get into trouble with each other. Time will tell and let him know who he is. In the face of your employee's behavior, you have reason to end up for the other party, but don't blame the other party, otherwise it will attract the other party's dissatisfaction and affect your future.