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]
(..) Oh, that was-- I don't even know what that's doing there. That's so weird. That was my fianc'e. - Your fianc'e? You were engaged. Yeah, a couple years ago. She's gone, though. She moved away. What happened? (..)2008 Two Lovers