Long Relational Algebraic expression provided than actually required
Trying to find a simpler solution is admirable, but your expression is subtly wrong.

The important part of the question is:


  ...or as a manager of the department...


Notice, that's not the same as "...or as a manager of the employee...".

Let's throw some names out there.


Ed White works for Brian Green. Ed is working on Project 123.
Project 123 is controlled by the Finance Department. Kathy Smith is the manager of the Finance Department.


So the requested result should include Project 123 in the output, because Project 123 is controlled by a department whose manager has a last name of Smith.

Your proposed solution will not include Project 123, because neither Ed nor his manager have a last name of Smith.