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Splitting U(0,0,1)F
U(a,b,c)F = aU(1,0,0)F + bU(0,1,0)F + cU(0,0,1)F.
The first two of the three 'basic series' are both the primus, the first one in reverse.
The third one, U(0,0,1)F, is not part of the fabfour and it's not an operator, but being the basic non-zero-series, its split is all the more interesting.
The splitvalues show that U(0,0,1)2, which is the 'the series of triangular numbers', cannot be split secundus- or tertius-based.
 U(0,0,1)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] -1 1/(F-2) 2/(F-2) 0 [2] 1 1/(F-2) F/(F-2) 1 [3] 0 -2/(F-2) -2/(F-2) 0 [4] 1 -F/(F-2) -2/(F-2) 0

 Splitting U(0,0,1)F Primusdoubled Quartusdoubled Secundusdoubled Tertius /(F-2)doubled Tertiusdoubled Secundus /(F-2)doubled Quartusdoubled Primusdoubled U-5 F4+F3-2F2-F+1 = -F * -F3-F2+2F+1 + 1 = F2-2 * (F3-F2-2F+1)/(F-2) + -F/(F-2) = F2-F-1 * (F3-3F)/(F-2) + -2/(F-2) = -F2-F+1 * -F2+1 + 0 U-4 F3+F2-F = -F * -F2-F+1 + 0 = F2-2 * (F2-F-1)/(F-2) + -2/(F-2) = F2-F-1 * (F2-2)/(F-2) + -2/(F-2) = -F2-F+1 * -F + 0 U-3 F2+F = -1 * -F2-F+1 + 1 = F * (F2-F-1)/(F-2) + -F/(F-2) = F-1 * (F2-2)/(F-2) + -2/(F-2) = -F-1 * -F + 0 U-2 F+1 = -1 * -F-1 + 0 = F * (F-1)/(F-2) + -2/(F-2) = F-1 * F/(F-2) + -2/(F-2) = -F-1 * -1 + 0 U-1 1 = 0 * -F-1 + 1 = 2 * (F-1)/(F-2) + -F/(F-2) = 1 * F/(F-2) + -2/(F-2) = -1 * -1 + 0 U0 0 = 0 * -1 + 0 = 2 * 1/(F-2) + -2/(F-2) = 1 * 2/(F-2) + -2/(F-2) = -1 * 0 + 0 U1 0 = 1 * -1 + 1 = F * 1/(F-2) + -F/(F-2) = 1 * 2/(F-2) + -2/(F-2) = 1 * 0 + 0 U2 1 = 1 * 1 + 0 = F * 1/(F-2) + -2/(F-2) = 1 * F/(F-2) + -2/(F-2) = 1 * 1 + 0 U3 F+1 = F * 1 + 1 = F2-2 * 1/(F-2) + -F/(F-2) = F-1 * F/(F-2) + -2/(F-2) = F+1 * 1 + 0 U4 F2+F = F * F+1 + 0 = F2-2 * (F-1)/(F-2) + -2/(F-2) = F-1 * (F2-2)/(F-2) + -2/(F-2) = F+1 * F + 0 U5 F3+F2-F = F2-1 * F+1 + 1 = F3-3F * (F-1)/(F-2) + -F/(F-2) = F2-F-1 * (F2-2)/(F-2) + -2/(F-2) = F2+F-1 * F + 0 U6 F4+F3-2F2-F+1 = F2-1 * F2+F-1 + 0 = F3-3F * (F2-F-1)/(F-2) + -2/(F-2) = F2-F-1 * (F3-3F)/(F-2) + -2/(F-2) = F2+F-1 * F2-1 + 0