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]
(..) Protocol . in effect. - We are at level, sir. - Good. Let's vanish. Engage retro-reflection panels. Reflection panels engaged. Gentlemen. Doctor, thank you for coming. Thanks for asking nicely. So, how long am I staying? Once we get our hands on (..)2012 The Avengers