Yes, really good question. When you write miller indices you are entitled to use any lattice vectors you like as long as you are clear about which ones you are using. For simple cubic, FCC, and BCC, it is conventional to use the edges of the cube as the lattice vectors. (These are not PLVs of the FCC or BCC lattice). This is related to the "important complication" in the notes. So while it is acceptable to choose other lattice vectors to work with, you would have to specify which ones you are using if they are not the conventional ones.
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