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The “mass-shell equation” E2 I3 = RE2 I3 + KE2 I3 = (RE I3 + i KE I3) x (RE I3 - i KE I3), where
KE I3 = [a12=a21=1= -a33, elsewhere 0, call Matrix X] x [b12=b21= cℏk = -b33, elsewhere 0], implies E = RE +/- i pc = RE + KE, where i p I3 = ℏk I3 = the 3 possible angular momenta, with k converting the (linear) meter into (circular) radians, i rotating the 3 possible planes, and a sign reversal engendering the anti-particle; also power expansion is not applicable here since v = c.
Set R3x3 ≡ [c12=1= -c23 = -c31, elsewhere 0]; then
R x Matrix X = [d11=d23=1 = -d32, elsewhere 0, call Matrix Z], and
– R2 x Matrix X = [e31=1= -e22 = -e13, elsewhere 0, call Matrix Y].
Projecting the above 3 matrices X, Z, Y onto (x,y),(x,z),(y,z) planes, we arrive at Pauli matrices. The first columns of the 3 matrices, being linear momenta, show semi-circular motions from W≡(-1,0,0) to N≡(0,1,0) to E≡(1,0,0) to T≡(0,0,1) and back to W, so electron has 4 states in 720°, as evidenced in Feynman III-11-1 with ℏ, not ℏ/2, and subject to Stern-Gerlach test.
