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]
(..) to work here with me, he got himself into a lot of trouble. I really miss the guy. Dr. Hodgins... You know, Angela? She and I, we were engaged to be married, and... That went south, too. That's a lot to bear... Especially if you feel responsible. Yeah, I do. (..)2008 TvShow: Bones Title: The Finger in the Nest Season: 4 Episode 3