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]
(..) (SHOUTING) (SIGHS) CHARLES: Even as we sit here, eating and drinking, Duke Rollo and our forces are engaged with the enemy. ROLAND: I know. And I have made every effort to make sure that, if the Duke should fail, Paris will still be well defended. (..)2016 TvShow: Vikings Title: The Last Ship Season: 4 Episode 10