Coordinate reflection bijection for tetracode lattices ($v\mapsto v-2v_i{e}_i$)

For the tetracode length-isospectral lattice construction $L_{\pm }=(\pm 3I+S){\mathbb{Z}}^4$, choose the first coordinate $i$ divisible by three and reverse its sign. Such a coordinate always exists because each nonzero tetracode word has one zero coordinate. The reflection preserves the reduction modulo three and switches the two lattice membership conditions modulo four. It is an involution between $L_{+}$ and $L_{-}$ preserving every absolute coordinate, so arbitrary positive diagonal weights preserve equality of their vector-length multisets.