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I will say in advance that I’m not 100% sure about these rotation axes. Here’s my attempt:
FE-I: the x-, y- and z-axes have 4-fold rotation symmetry. Therefore the EFG will be zero.
FE-II: the z-axis has 4-fold rotation symmetry and the x- and y-axes only 2-fold symmetry. Therefore I think all axes can be chosen as the z-axis, but eta is zero only if the fourfold axis is chosen.
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