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]
(..) It would introduce an unstable element to a critical situation. Your experience makes you the perfect man to lead this fight. Bridge to Captain Picard. We've received word from the fleet, they've engaged the Borg. Mr Data, put Starfleet frequency on audio. Flagship Endeavour, stand by to engage. (..)1996 Star Trek: First Contact