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]
(..) Go on please. She lived with me in Versailles. Worked at the Institute. What Institute? The institute named after Stanislas Boudriard. It's engaged in the socialization of the blind youth. Is she blind? Yes. From birth? Yes. (..)1981 To Kill a Cop
(..) wanted to come earlier... Earlier than who? Pigot. The one who's in your office now. I tried to call you It was engaged all the time I was out to drink a cup of coffee. You're fired! From now on or shall we see later about that? We'll see. Shall we talk? (..)1981 To Kill a Cop