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]
(..) Outer Party member Davies, Miniprod, Women's Section. That concludes the list of known agents of Goldstein actively engaged in the latest conspiracy... Not much of this gets to the Outer Party, I'm afraid. To our leader. Our eternal allies in Eurasia are turning against us. - You can...? - Yes. (..)1984 1984