1. The first way to determine the presence of axial symmetry in the electric field gradient is to look at the toy model describing the system. The model obviously possesses axial symmetries (the number of which depends on the orientation of the dumbbell-shaped nucleus). One of these would be along the x-axis coming out of the plane of the paper, another one aligned with the direction of the dumbbell.
2. The second is by considering the formula for the perturbing hamiltonian given on the slides. I assume that if we can show that there is an axial symmetry in the Hamiltonian, the axial symmetry of the EFG tensor follows.
Lets assume we pick the z axis as our axis to check for symmetry. This means that the x and y (I_x and I_y in this case?) values in the Hamilonian have to be replaced with -x and -y respectively. If we find the same Hamiltonian, we have and axial symmetry.
However, applied to the Hamiltonian given, we run into a problem with the 4I(2I-1)hbar term (if I represents the vector I, not the scalar value of I), so I’m assuming I made a mistake somewhere (or I is a scalar in which case it should all work out).