= Solution
For type I, a <magnetic unit cell> uses the magnetic Bravais translations $(0,a,a)$, $(a,0,a)$ and $(a,a,0)$. They form an <face-centered cubic lattice>[FCC lattice] with conventional cubic parameter $2a$. A two-<spin> <crystal motif>[basis] is an up <spin> at $(0,0,0)$ and a down <spin> at $(a,0,0)$, or fractional conventional coordinates $(0,0,0)$ and $(1/2,0,0)$. Every translation preserves the parity of $n_x+n_y+n_z$.
At $\mathbf Q=(2\pi/a)(h,k,l)$, the FCC translation sum allows $h,k,l$ all integers or all half-odd-integers. The <spin> <crystal motif>[basis] contributes $S[1-e^{2\pi ih}]$, which removes the integer set and leaves
$$
\boxed{\text{type I: }(h,k,l)=(n+\tfrac12,m+\tfrac12,p+\tfrac12).}
$$
Equivalently the <spin> field is $S(-1)^{n_x+n_y+n_z}$, so <magnetic Bragg peaks> are nuclear reciprocal vectors shifted by $(1/2,1/2,1/2)$.
For type II with parallel <spins> along $z$, use $(a,a,0)$, $(-a,a,0)$ and $(0,0,a)$ as magnetic translations: a primitive tetragonal cell with side lengths $\sqrt2a,\sqrt2a,a$, containing an up <spin> at zero and a down <spin> at $(a,0,0)$. The reciprocal translations have $h,k$ both integers or both half-odd-integers, with $l$ integer. The same <crystal motif>[basis] factor removes the integer-$h$ set:
$$
\boxed{\text{type II, parallel }z:\ (h,k,l)=(n+\tfrac12,m+\tfrac12,p).}
$$
This also follows from $S(-1)^{n_x+n_y}$. Other domains permute the three coordinate directions. A magnetic <neutron> cross section further multiplies these <Bravais lattice>[lattice] conditions by the <magnetic form factor> and by the component of ordered moment perpendicular to $\mathbf Q$; that polarization factor can suppress an otherwise allowed peak.
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