He won't be able to attend because he's 'otherwise engaged'. [=he is busy doing something else]
('US') He is 'engaged' in research. = ('Brit') He is 'engaged' on research.
She 'engaged' him 'in' conversation. [=she started a conversation with him]
(..) reports of a metal man, or men, coming in and emptying the place. - Fatalities? - Only when engaged. Mostly guys left in a fugue state going on about old memories, worst fears, and something too fast to see. Maximoffs. Well, that makes sense he'd go to them, they have someone in common. (..)2015 Avengers: Age of Ultron