Once upon a time presidents would follow the rules by scheduling a visit/inspection of a military base or government office in the location where the vacation would take place, and do that on the way.
That let them use AF1 on the way, and the vacation would officially start after the end of the vacation. Don’t know if the current administration does that or not.
OK, but is Mrs. Obama the Second In Command of the military then? A visit isn’t an inspection. The first inspection I stood, the Captain flipped a silver dollar on my bunk. It didn’t bounce high enough. He tipped over my bunk and my wall locker and told me I was descended from dogs. Mrs. Obama can be stern, but I don’t think she’d do that to anyone.