First, it should be called Face Centered Orthorhombic, not FCC orthorhombic, since the last "C" stands for cubic.
The calculation should be entirely analogous to the cubic case. You can write the PLVs of the FC orthorhombic (the "a"'s) then use the formula to find a set of reciprocal PLVs (the "b"'s). You should be able to reassemble (take linear combinations) of these PLVs to convince yourself that the b's now describe a Body centered orthorhombic.
I will work through it again to try to see if I can figure out what the pitfalls are. I might discuss this one in revision lectures.
For 6.13 this is supposed to be identical to the antiferromagnet problem. But I guess it seems a bit confusing. I may discuss this in revision lectures.