Working hard for someone who beat you for the role takes more character than winning ever did.

Losing a position you wanted leaves you with a choice nobody will ever ask you about directly. You can withhold a little, in ways that are easy to justify and impossible to prove, or you can give the same effort you would have wanted from the people under you. Only one of those leaves you liking yourself in a year.

Decide early rather than in the moment, because the moment is when the small withholdings happen. Back the person who got it in public, and take your disappointment somewhere private and real. You are allowed to be hurt about it and still be useful.