You have to look carefully at the structure of the crystal. Recall there are two atoms in the basis --- one at  and one at [1/4,1/4,1/4]. In the (111) direction, if you start at the atom at position , you move [1/4,1/4,1/4] which is in the (111) direction to get to the neighbor atom. However, if you start at the [1/4,1/4,1/4] and move in the (111) direction, there is no atom there! The bonds are at some diagonal angle (and there are three of them!)--- while the bonds are actually all the same length, it is clear that this effective spring constant must be different from the one between  and [1/4,1/4,1/4].
Does that help?
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