  In Parts I and II we showed that e, $\nu$ propagators can be derived from
rotation invariant projection operators, thereby providing examples of how
quantities with spacetime symmetry can be obtained by constraining rotationally
symmetric objects. One constraint is the restriction of the basis; only two
kinds of bases were considered, one for the electron and one for the neutrino.
In this part, we find that, of a wide range of bases each consistent with the
constraint process, only the two kinds of bases considered in Parts I and II
give spacetime symmetric propagators. We interpret the result geometrically.
The spinor representation is unfaithful in four dimensional Euclidean space
which explains why spin 1/2 wave functions have four, not two, components. Then
we show how a basis relates to two planes in four dimensional Euclidean space.
A pair of planes spanning two or three dimensions does not allow spacetime
symmetry. Spacetime symmetry requires two planes that span four dimensions.
  PACS: 11.30.-j, 11.30.Cp, and 03.65.Fd
