 Solution

The FCC parity rule forbids $(100)$ and $(110)$ and allows $(200)$ and $(111)$. To follow each same plane family through the deformation, use fractional coordinates $(x,y,z)$ in the compressed old cell and $(X,Y,Z)$ in the new cube. The preceding vectors giveThus $hx+ky+lz=m$ becomes $HX+KY+LZ=m$, withCrucially, this transformation retains the plane spacing and diffraction order, not just a normal direction:

The two physically allowed reflections remain allowed, and the two absent ones remain absent. Their angles and spacings change under strain; their allowedness as tracked reciprocal vectors does not.

The half-integer entry is deliberate. It cannot be relabeled as the ordinary BCC $(1\bar{1}0)$ reflection while claiming to track the same diffraction order. Multiplying its indices by two doubles the reciprocal vector and halves the spacing; that new allowed reflection is the image of old FCC $(200)$, not old $(100)$. If only an orientation label is wanted, the old $(100)$ and $(200)$ planes both have the new $(1\bar{1}0)$ orientation, but are distinct orders. This is the distinction between diffraction indexing under a cell change and merely reducing a plane normal to the smallest integer ratio.