= Solution
Ignoring the common normalization $1/(2\sqrt2)$, direct multiplication gives
$$
|v(e_{00})\rangle
=|000\rangle+|001\rangle+|010\rangle-|011\rangle,
$$
$$
|v(e_{10})\rangle
=|000\rangle-|001\rangle+|010\rangle+|011\rangle,
$$
$$
|v(e_{01})\rangle
=|100\rangle+|101\rangle-|110\rangle+|111\rangle,
$$
$$
|v(e_{11})\rangle
=|100\rangle-|101\rangle-|110\rangle-|111\rangle.
$$
These four states span the positive eigenspace of $Z\otimes X\otimes Z$. The parent term annihilating that local MPS support is therefore the complementary projector
$$
\boxed{h=\frac12(I-Z\otimes X\otimes Z)}.
$$
Translated terms have stabilizers $K_j=Z_{j-1}X_jZ_{j+1}$. Two such Pauli strings either do not overlap nontrivially or have two X--Z anticommutations, so all $K_j$ commute. Their positive eigenspaces have a common state, making the parent Hamiltonian frustration free. Each alternating global X symmetry flips the two Z factors of every $K_j$ or neither, and therefore commutes with every local term.
Solved by gpt-5.6-sol high.
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