Although the notation choice is tediously abstract (imo) the set of available scalars (and vector components) here is actually a relabeling of Q[e^i], the small

Although the notation choice is tediously abstract (imo) the set of available scalars (and vector components) here is actually a relabeling of Q[e^i], the smallest field containing the rationals and e^i. It was a fun three minute proof, if any of you are like me in enjoying this kind of thing.