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]
(..) about an ogre problem? You are told correct. But for this, I charge a great deal of money. Would... this be enough? You have engaged my valuable services, Your Majesty. Just tell me where I can find this ogre. # Everyone says I'm getting down too low (..)2004 Shrek 2