Crystallographic information can help you to decide which components of a force vector must be zero. That’s less left to compute. You will be able to use Wyckoff tables to determine in which directions symmetry allows an atom to ‘move’.
Take your orthorhombic Fe-Al cell. It has space group number 59 (Pmmn), which is something you could have found yourself by applying what you learned in the chapter on crystallography. Use this space group information to find which components of the internal position coordinates in our Fe-Al are not fixed by symmetry.