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]
(..) - Nichols, we gotta stop. - No, we can't. I gotta work out the other side. Otherwise I'll be asymmetrical like a crab, or a tennis pro. No, no, no. No more. I'm engaged. Yeah, that's right. Get it while you can. No! No, it's not fair to Christopher. And I need to start tightening up. (..)2013 TvShow: Orange Is the New Black Title: The Chickening Season: 1 Episode 5