Yes, actually this turns out to be the same as the solution based on the invariant taking values in the Klein-four group: the kernel of the map f:Z3→(Z/2Z)2 given by (x1,x2,x3)→(x1+x2,x1+x3) is precisely the subspace of integer vectors with all entries having the same parity. It’s just that the invariant solution makes it look natural, while doing it this way around leads to having to express these parity conditions in a more elaborate way.
Yes, actually this turns out to be the same as the solution based on the invariant taking values in the Klein-four group: the kernel of the map f:Z3→(Z/2Z)2 given by (x1,x2,x3)→(x1+x2,x1+x3) is precisely the subspace of integer vectors with all entries having the same parity. It’s just that the invariant solution makes it look natural, while doing it this way around leads to having to express these parity conditions in a more elaborate way.