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]
(..) Penny and I got into an argument about George Washington. So, I made him show me the WABAC. And I lost her in ancient Egypt. And I got engaged to King Tut. So, I came back and got you. Then we ran out of gas. In Florence. Went into a black hole. And then you died (..)2014 Mr. Peabody & Sherman