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]
(..) this line of thought will carry us. Hmm. Great care we must take. Chancellor We’ve just received a report from Master Kenobi. He has engaged General Grievous. We can only hope that Master Kenobi is up to the challenge. I should be there with him. It’s upsetting to me to see that the council... (..)2005 Star Wars: Episode III - Revenge of the Sith