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]
(..) Yet what can it, when one cannot repent? Oh wretched state, oh bosom black as death Oh limed soul, that struggling to be free an more engaged Help, angels, make assay. Bow, stubborn knees And heart with strings of steel, be soft as sinews of the new-born babe All may be well (..)2015 Hamlet