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]
(..) Para un americano no habla mal espanol Really? I'm sure you speak English very well, too He says they've engaged the Sandinistas regularly He expects shooting tonight If anything happens to me tell every woman I've ever dated (..)1987 Broadcast News