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]
(..) When you grow up, if you still feel raw about it... I'll be waiting. For those regarded as warriors... When engaged in combat... the vanquishing of thine enemy can be the warrior's only concern. Suppress a human emotion and compassion... ...kill whoever stands in thy way, even if that be (..)2003 Kill Bill: Vol. 1