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]
(..) wants its gold back. Your father never minced words either. But, yes, your debts are considerable. And you're now engaged in a conflict on several fronts. We both know how expensive war can be. And we both know gold wins wars. Your vaults are empty. (..)2017 TvShow: Game of Thrones Title: The Queen's Justice Season: 7 Episode 3