 Rationally independent axis weights obstruct a lattice isometry

Choose positive axis weights linearly independent over the rationals, and suppose both integer Euclidean lattices contain $q{\mathbb{Z}}^d$ for some positive integer $q$. A length-preserving linear map sends each $q{e}_i$ to an integer vector with exactly that absolute-coordinate pattern, by the displayed implication. Thus the map must be a diagonal sign change. For the four-coordinate tetracode, preserving the code forces all four signs to agree, because each of its four nonzero projective words has support on a different triple of coordinates. A global sign cannot exchange the two odd-vector parity conditions, proving nonisometry for these weights.