Authority becomes legitimacy when people believe the rules apply upward too.

Legitimacy comes from the rules applying upward. People will accept a great deal of constraint if they can see it binding the people who wrote it, and they will resist even reasonable rules that visibly stop at a certain level. What is being judged is not the rule but whether it is real.

Where you set standards, make your own compliance visible. Follow the tedious ones publicly. It costs little and it is the entire foundation of anyone taking the rest seriously.