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This curious phenomenom requires an example first.
Consider U(-2,3,-5)7 as representative of all U(a,b,c)F.
What is the relationship between this non-zero series, and the following pairs of zero series?
• U(0,1)7 - the primus of 7 - and U(25,20)7
• U(2,7)7 - the secundus of 7 - and U(-3,2)7
• U(1,1)7 - the tertius of 7 - and U(-1,19)7
• U(-1,1)7 - the quartus of 7 - and U(5,15)7
It is this:

 U(-2,3,-5)7 Primus 7doubled U(25,20)7doubled Secundus 7doubled U(-3,2)7doubled Tertius 7doubled U(-1,19)7doubled Quartus 7doubled U(5,15)7doubled U-9 -112287722 = -329 * 341300 + -22 = 2207 * -50878 + 24 = 1926 * -58301 + 4 = -2584 * 43455 + -2 U-8 -16382557 = -329 * 49795 + -2 = 2207 * -7423 + 4 = 1926 * -8506 + -1 = -2584 * 6340 + 3 U-7 -2390182 = -48 * 49795 + -22 = 322 * -7423 + 24 = 281 * -8506 + 4 = -377 * 6340 + -2 U-6 -348722 = -48 * 7265 + -2 = 322 * -1083 + 4 = 281 * -1241 + -1 = -377 * 925 + 3 U-5 -50877 = -7 * 7265 + -22 = 47 * -1083 + 24 = 41 * -1241 + 4 = -55 * 925 + -2 U-4 -7422 = -7 * 1060 + -2 = 47 * -158 + 4 = 41 * -181 + -1 = -55 * 135 + 3 U-3 -1082 = -1 * 1060 + -22 = 7 * -158 + 24 = 6 * -181 + 4 = -8 * 135 + -2 U-2 -157 = -1 * 155 + -2 = 7 * -23 + 4 = 6 * -26 + -1 = -8 * 20 + 3 U-1 -22 = 0 * 155 + -22 = 2 * -23 + 24 = 1 * -26 + 4 = -1 * 20 + -2 U0 -2 = 0 * 25 + -2 = 2 * -3 + 4 = 1 * -1 + -1 = -1 * 5 + 3 U1 3 = 1 * 25 + -22 = 7 * -3 + 24 = 1 * -1 + 4 = 1 * 5 + -2 U2 18 = 1 * 20 + -2 = 7 * 2 + 4 = 1 * 19 + -1 = 1 * 15 + 3 U3 118 = 7 * 20 + -22 = 47 * 2 + 24 = 6 * 19 + 4 = 8 * 15 + -2 U4 803 = 7 * 115 + -2 = 47 * 17 + 4 = 6 * 134 + -1 = 8 * 100 + 3 U5 5498 = 48 * 115 + -22 = 322 * 17 + 24 = 41 * 134 + 4 = 55 * 100 + -2 U6 37678 = 48 * 785 + -2 = 322 * 117 + 4 = 41 * 919 + -1 = 55 * 685 + 3 U7 258243 = 329 * 785 + -22 = 2207 * 117 + 24 = 281 * 919 + 4 = 377 * 685 + -2 U8 1770018 = 329 * 5380 + -2 = 2207 * 802 + 4 = 281 * 6299 + -1 = 377 * 4695 + 3 U9 12131878 = 2255 * 5380 + -22 = 15127 * 802 + 24 = 1926 * 6299 + 4 = 2584 * 4695 + -2

The curious thing is that a non-zero series is split, in four different ways, in a peculiar 'product' of two zero series, called the operator, which is always one of the fabfour, and the output series.
In each split there are two constants involved, but they are outside the multiplication with the factor F, whereas the constant '-5' in the original series operates inside the multiplication.

Compare U(-2,3,-5)7 term by term with U(-2,3)7 to see its influence.
Note that the constant slows down the series in one direction, while accelerating it in the other.
 U(-2,3)7 -37677 -5497 -802 -117 -17 -2 3 23 158 1083 7423 50878 U(-2,3,-5)7 -50877 -7422 -1082 -157 -22 -2 3 18 118 803 5498 37678

Yet this deviation created by the constant is somehow 'overcome' in the above splits.

Is this a freak example?
Unfortunately not. I can split any series U(a,b,c)F in the above four ways, using the fabfour as operators. So can you once you know the right way to put the primus-, secundus-, tertius- and quartus-splitvalues into the 'core' of the splittable, formed by the four rows U0 - U3.

Splitvalues?
Sixteen in total - four of them for each of the fabfour. They were not arrived at by any logic or reasoning. They were arrived at by painstakingly digging through multitudes of examples, in part found randomly in the course of other investigations, in part created for the very purpose of finding the patterns among the constantly shifting parameters.
It took me a couple of months in which I dug and kept digging, and calcguessing, and eventually these values emerged - they are what they are, what they were before God opened his eyes, what they will be after he shuts his mouth (which doesn't appear to be any time soon).
 U(a,b,c)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] -Fa+2b-c a+ c/(F-2) (a+b)+ 2c/(F-2) -(a-b) [2] Fb-2a+c b+ c/(F-2) Fb-(a-b)+ Fc/(F-2) Fb-(a+b)+c [3] a -a- 2c/(F-2) -b- 2c/(F-2) b [4] Fa-b+c -Fa+b- Fc/(F-2) -a- 2c/(F-2) a

I don't know how or why they work. They are not derived from anything, nor does the operation they make possible relate to anything known in elementary number theory. An enigma.

How does it work?
Consider the example. For U(-2,3,-5)7 the splitvalues are:
 U(-2,3,-5)7 Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] 25 -3 -1 5 [2] 20 2 19 15 [3] -2 4 -1 3 [4] -22 24 4 -2

That wasn't too hard. Now place these values in exactly the above positions in the core of the splittable - the rows U0-U3 - and develop the series involved accordingly. That would be all, thank you.
The equalities will keep on track ad infinitum regardless of the choice of a, b, c and F.

What if (F-2) doesn't divide 'c'?
Then you will encounter fractions in the secundus' en tertius' splitvalues and consequently in the output series. Secundus- en tertius-based splits therefore aren't closed but jump into the U(p/r, q/r)F domain. It doesn't affect the equalities - they will still keep on track ad infinitum.

What if F=2?
Then, unless c=0, there are no secundus- and tertius-based splits possible. U(2,2)2, the secundus of 2, is a series of 'two's' while U(1,1)2, the tertius of 2, is a series of 'one's'.

What if F=2 and c=0?
Then the train remains on track: zero series can be split for F=2.

Splitting U(0,1)F
 U(0,1)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] 2 0 1 1 [2] F 1 F+1 F-1 [3] 0 0 -1 1 [4] -1 1 0 0

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

Note that the secundus-based split of a zero-series is neutral: what goes in comes out - in this case the primus.
Note also that a series operating on itself - in this case the primus-based split - renders the secundus as its output series.

Splitting U(2,F)F
 U(2,F)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] 0 2 F+2 F-2 [2] F2-4 F F2+F-2 F2-F-2 [3] 2 -2 -F F [4] F -F -2 2

 Splitting U(2,F)Fthe secundus Primusdoubled Primus*(F2-4)doubled Secundusdoubled Secundusdoubled Tertiusdoubled Tertius*(F+2)doubled Quartusdoubled Quartus*(F-2)doubled U-5 F5-5F3+5F = -F * -(F2-1)*(F2-4) + F = F2-2 * F3-3F + -F = F2-F-1 * (F2-F-1)*(F+2) + -2 = -F2-F+1 * -(F2+F-1)*(F-2) + 2 U-4 F4-4F2+2 = -F * -F*(F2-4) + 2 = F2-2 * F2-2 + -2 = F2-F-1 * (F-1)*(F+2) + -F = -F2-F+1 * -(F+1)*(F-2) + F U-3 F3-3F = -1 * -F*(F2-4) + F = F * F2-2 + -F = F-1 * (F-1)*(F+2) + -2 = -F-1 * -(F+1)*(F-2) + 2 U-2 F2-2 = -1 * -(F2-4) + 2 = F * F + -2 = F-1 * F+2 + -F = -F-1 * -(F-2) + F U-1 F = 0 * -(F2-4) + F = 2 * F + -F = 1 * F+2 + -2 = -1 * -(F-2) + 2 U0 2 = 0 * 0 + 2 = 2 * 2 + -2 = 1 * F+2 + -F = -1 * F-2 + F U1 F = 1 * 0 + F = F * 2 + -F = 1 * F+2 + -2 = 1 * F-2 + 2 U2 F2-2 = 1 * F2-4 + 2 = F * F + -2 = 1 * (F-1)*(F+2) + -F = 1 * (F+1)*(F-2) + F U3 F3-3F = F * F2-4 + F = F2-2 * F + -F = F-1 * (F-1)*(F+2) + -2 = F+1 * (F+1)*(F-2) + 2 U4 F4-4F2+2 = F * F*(F2-4) + 2 = F2-2 * F2-2 + -2 = F-1 * (F2-F-1)*(F+2) + -F = F+1 * (F2+F-1)*(F-2) + F U5 F5-5F3+5F = F2-1 * F*(F2-4) + F = F3-3F * F2-2 + -F = F2-F-1 * (F2-F-1)*(F+2) + -2 = F2+F-1 * (F2+F-1)*(F-2) + 2

Note that the secundus-based split of a zero-series is neutral: what goes in comes out - in this case the secundus itself.
Note also that a series operating on itself - in this case also the secundus - renders the secundus as its output series.
All other splits of the secundus give a multiple of the operator as output.

Splitting U(1,1)F
 U(1,1)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] -F+2 1 2 0 [2] F-2 1 F F-2 [3] 1 -1 -1 1 [4] F-1 -F+1 -1 1

 Splitting U(1,1)Fthe tertius Primusdoubled Quartus*(F-2)doubled Secundusdoubled Tertiusdoubled Tertiusdoubled Secundusdoubled Quartusdoubled Primus*(F-2)doubled U-5 F5-F4-4F3+3F2+3F-1 = -F * -(F3+F2-2F-1)*(F-2) + F-1 = F2-2 * F3-F2-2F+1 + -F+1 = F2-F-1 * F3-3F + -1 = -F2-F+1 * -(F2-1)*(F-2) + 1 U-4 F4-F3-3F2+2F+1 = -F * -(F2+F-1)*(F-2) + 1 = F2-2 * F2-F-1 + -1 = F2-F-1 * F2-2 + -1 = -F2-F+1 * -F*(F-2) + 1 U-3 F3-F2-2F+1 = -1 * -(F2+F-1)*(F-2) + F-1 = F * F2-F-1 + -F+1 = F-1 * F2-2 + -1 = -F-1 * -F*(F-2) + 1 U-2 F2-F-1 = -1 * -(F+1)*(F-2) + 1 = F * F-1 + -1 = F-1 * F + -1 = -F-1 * -(F-2) + 1 U-1 F-1 = 0 * -(F+1)*(F-2) + F-1 = 2 * F-1 + -F+1 = 1 * F + -1 = -1 * -(F-2) + 1 U0 1 = 0 * -(F-2) + 1 = 2 * 1 + -1 = 1 * 2 + -1 = -1 * 0 + 1 U1 1 = 1 * -(F-2) + F-1 = F * 1 + -F+1 = 1 * 2 + -1 = 1 * 0 + 1 U2 F-1 = 1 * F-2 + 1 = F * 1 + -1 = 1 * F + -1 = 1 * F-2 + 1 U3 F2-F-1 = F * F-2 + F-1 = F2-2 * 1 + -F+1 = F-1 * F + -1 = F+1 * F-2 + 1 U4 F3-F2-2F+1 = F * (F+1)*(F-2) + 1 = F2-2 * F-1 + -1 = F-1 * F2-2 + -1 = F+1 * F*(F-2) + 1 U5 F4-F3-3F2+2F+1 = F2-1 * (F+1)*(F-2) + F-1 = F3-3F * F-1 + -F+1 = F2-F-1 * F2-2 + -1 = F2+F-1 * F*(F-2) + 1

Note that the secundus-based split of a zero-series is neutral: what goes in comes out - in this case the tertius.
Note also that a series operating on itself - in this case the tertius - renders the secundus as its output series.

Splitting U(-1,1)F
 U(-1,1)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] F+2 -1 0 2 [2] F+2 1 F+2 F [3] -1 1 -1 1 [4] -F-1 F+1 1 -1

 Splitting U(-1,1)Fthe quartus Primusdoubled Tertius*(F+2)doubled Secundusdoubled Quartusdoubled Tertiusdoubled Primus*(F+2)doubled Quartusdoubled Secundusdoubled U-5 -F5-F4+4F3+3F2-3F-1 = -F * (F3-F2-2F+1)*(F+2) + -F-1 = F2-2 * -F3-F2+2F+1 + F+1 = F2-F-1 * -(F2-1)*(F+2) + 1 = -F2-F+1 * F3-3F + -1 U-4 -F4-F3+3F2+2F-1 = -F * (F2-F-1)*(F+2) + -1 = F2-2 * -F2-F+1 + 1 = F2-F-1 * -F*(F+2) + -1 = -F2-F+1 * F2-2 + 1 U-3 -F3-F2+2F+1 = -1 * (F2-F-1)*(F+2) + -F-1 = F * -F2-F+1 + F+1 = F-1 * -F*(F+2) + 1 = -F-1 * F2-2 + -1 U-2 -F2-F+1 = -1 * (F-1)*(F+2) + -1 = F * -F-1 + 1 = F-1 * -(F+2) + -1 = -F-1 * F + 1 U-1 -F-1 = 0 * (F-1)*(F+2) + -F-1 = 2 * -F-1 + F+1 = 1 * -(F+2) + 1 = -1 * F + -1 U0 -1 = 0 * F+2 + -1 = 2 * -1 + 1 = 1 * 0 + -1 = -1 * 2 + 1 U1 1 = 1 * F+2 + -F-1 = F * -1 + F+1 = 1 * 0 + 1 = 1 * 2 + -1 U2 F+1 = 1 * F+2 + -1 = F * 1 + 1 = 1 * F+2 + -1 = 1 * F + 1 U3 F2+F-1 = F * F+2 + -F-1 = F2-2 * 1 + F+1 = F-1 * F+2 + 1 = F+1 * F + -1 U4 F3+F2-2F-1 = F * (F-1)*(F+2) + -1 = F2-2 * F+1 + 1 = F-1 * F*(F+2) + -1 = F+1 * F2-2 + 1 U5 F4+F3-3F2-2F+1 = F2-1 * (F-1)*(F+2) + -F-1 = F3-3F * F+1 + F+1 = F2-F-1 * F*(F+2) + 1 = F2+F-1 * F2-2 + -1

Note that the secundus-based split of a zero-series is neutral: what goes in comes out - in this case the quartus.
Note also that a series operating on itself - in this case the quartus - renders the secundus as its output series.

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

Splitting U(a,b)F
 U(a,b)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] -Fa+2b a a+b -(a-b) [2] -2a+Fb b Fb-(a-b) Fb-(a+b) [3] a -a -b b [4] Fa-b -Fa+b -a a

 Splitting U(a,b)Fthe general zero series Primusdoubled U(-Fa+2b, Fb-2a)Fdoubled Secundusdoubled U(a,b)Fdoubled Tertiusdoubled U[a+b, Fb-(a-b)]Fdoubled Quartusdoubled U[-(a-b), Fb-(a+b)]Fdoubled U-4 F4a-F3b-3F2a+2Fb+a = -F * -F3a+F2b+3Fa-2b + a = F2-2 * F2a-Fb-a + -a = F2-F-1 * F2a+F(a-b)-(a+b) + -b = -F2-F+1 * -F2a+F(a+b)+(a-b) + b U-3 F3a-F2b-2Fa+b = -1 * -F3a+F2b+3Fa-2b + Fa-b = F * F2a-Fb-a + -Fa+b = F-1 * F2a+F(a-b)-(a+b) + -a = -F-1 * -F2a+F(a+b)+(a-b) + a U-2 F2a-Fb-a = -1 * -F2a+Fb+2a + a = F * Fa-b + -a = F-1 * Fa+(a-b) + -b = -F-1 * -Fa+(a+b) + b U-1 Fa-b = 0 * -F2a+Fb+2a + Fa-b = 2 * Fa-b + -Fa+b = 1 * Fa+(a-b) + -a = -1 * -Fa+(a+b) + a U0 a = 0 * -Fa+2b + a = 2 * a + -a = 1 * a+b + -b = -1 * -(a-b) + b U1 b = 1 * -Fa+2b + Fa-b = F * a + -Fa+b = 1 * a+b + -a = 1 * -(a-b) + a U2 Fb-a = 1 * Fb-2a + a = F * b + -a = 1 * Fb-(a-b) + -b = 1 * Fb-(a+b) + b U3 F2b-Fa-b = F * Fb-2a + Fa-b = F2-2 * b + -Fa+b = F-1 * Fb-(a-b) + -a = F+1 * Fb-(a+b) + a U4 F3b-F2a-2Fb+a = F * F2b-Fa-2b + a = F2-2 * Fb-a + -a = F-1 * F2b-F(a-b)-(a+b) + -b = F+1 * F2b-F(a+b)+(a-b) + b U5 F4b-F3a-3F2b+2Fa+b = F2-1 * F2b-Fa-2b + Fa-b = F3-3F * Fb-a + -Fa+b = F2-F-1 * F2b-F(a-b)-(a+b) + -a = F2+F-1 * F2b-F(a+b)+(a-b) + a

Note that the secundus-based split of a zero-series is neutral: what goes in comes out - in this case the general zero-series itself.

Splitting U(a,b,c)F
 U(a,b,c)F Primus splitvalues Secundus splitvalues Tertius splitvalues Quartus splitvalues [1] -Fa+2b-c a+ c/(F-2) (a+b)+ 2c/(F-2) -(a-b) [2] Fb-2a+c b+ c/(F-2) Fb-(a-b)+ Fc/(F-2) Fb-(a+b)+c [3] a -a- 2c/(F-2) -b- 2c/(F-2) b [4] Fa-b+c -Fa+b- Fc/(F-2) -a- 2c/(F-2) a

 Splitting U(a,b,c)Fthe general series Primusdoubled U(-Fa+2b-c, Fb-2a+c)Fdoubled Secundusdoubled U[a+ c/(F-2), b+ c/(F-2)]Fdoubled Tertiusdoubled U[a+b+ 2c/(F-2), Fb-(a-b)+ Fc/(F-2)]Fdoubled Quartusdoubled U[-(a-b), Fb-(a+b)+c]Fdoubled U-4 F4a-F3(b-c)-F2(3a-c)+F(2b-c)+a = -F * -F3a+F2(b-c)+F(3a-c)-2b+c + a = F2-2 * F2a-Fb-a+ c(F2-F-1)/(F-2) + -a- 2c/(F-2) = F2-F-1 * F2a+F(a-b)-(a+b) +c(F2-2)/(F-2) + -b -2c/(F-2) = -F2-F+1 * -F2a+F(a+b-c)+(a-b) + b U-3 F3a-F2(b-c)-F(2a-c)+b = -1 * -F3a+F2(b-c)+F(3a-c)-2b+c + Fa-b+c = F * F2a-Fb-a+ c(F2-F-1)/(F-2) + -Fa+b -Fc/(F-2) = F-1 * F2a+F(a-b)-(a+b) +c(F2-2)/(F-2) + -a -2c/(F-2) = -F-1 * -F2a+F(a+b-c)+(a-b) + a U-2 F2a-F(b-c)-(a-c) = -1 * -F2a+F(b-c)+2a-c + a = F * Fa-b+ c(F-1)/(F-2) + -a- 2c/(F-2) = F-1 * Fa+(a-b) +Fc/(F-2) + -b -2c/(F-2) = -F-1 * -Fa+(a+b)-c + b U-1 Fa-(b-c) = 0 * -F2a+F(b-c)+2a-c + Fa-b+c = 2 * Fa-b+ c(F-1)/(F-2) + -Fa+b -Fc/(F-2) = 1 * Fa+(a-b) +Fc/(F-2) + -a -2c/(F-2) = -1 * -Fa+(a+b)-c + a U0 a = 0 * -Fa+2b-c + a = 2 * a+ c/(F-2) + -a- 2c/(F-2) = 1 * (a+b)+ 2c/(F-2) + -b -2c/(F-2) = -1 * -(a-b) + b U1 b = 1 * -Fa+2b-c + Fa-b+c = F * a+ c/(F-2) + -Fa+b -Fc/(F-2) = 1 * (a+b)+ 2c/(F-2) + -a -2c/(F-2) = 1 * -(a-b) + a U2 Fb-(a-c) = 1 * Fb-2a+c + a = F * b+ c/(F-2) + -a- 2c/(F-2) = 1 * Fb-(a-b)+ Fc/(F-2) + -b -2c/(F-2) = 1 * Fb-(a+b)+c + b U3 F2b-F(a-c)-(b-c) = F * Fb-2a+c + Fa-b+c = F2-2 * b+ c/(F-2) + -Fa+b -Fc/(F-2) = F-1 * Fb-(a-b)+ Fc/(F-2) + -a -2c/(F-2) = F+1 * Fb-(a+b)+c + a U4 F3b-F2(a-c)-F(2b-c)+a = F * F2b-F(a-c)-2b+c + a = F2-2 * Fb-a+ c(F-1)/(F-2) + -a- 2c/(F-2) = F-1 * F2b-F(a-b)-(a+b)+ c(F2-2)/(F-2) + -b -2c/(F-2) = F+1 * F2b-F(a+b-c)+(a-b) + b U5 F4b-F3(a-c)-F2(3b-c)+F(2a-c)+b = F2-1 * F2b-F(a-c)-2b+c + Fa-b+c = F3-3F * Fb-a+ c(F-1)/(F-2) + -Fa+b -Fc/(F-2) = F2-F-1 * F2b-F(a-b)-(a+b)+ c(F2-2)/(F-2) + -a -2c/(F-2) = F2+F-1 * F2b-F(a+b-c)+(a-b) + a

This is the alpha and omega of splitting in the U(a,b,c)F domain: the split of the general series.
Secundus- and tertius-based splits go clearly into the realm of rational numbers. Clearly too, the secundus loses the 'unity role' it has for zero series.

Matrices
 Matrix of the output-series of splits of the fabfour and U(0,0,1)F Primus based Secundus based Tertius based Quartus based Primus secundus primus quartus tertius Secundus primus*(F+2)(F-2) secundus tertius*(F+2) quartus*(F-2) Tertius quartus*(F-2) tertius secundus primus*(F-2) Quartus tertius*(F+2) quartus primus*(F+2) secundus U(0,0,1)F quartus tertius/(F-2) secundus/(F-2) primus

The outputseries in the blue area have an index-shift of two with regard to the corresponding outputseries in the yellow area (and with the rest of the matrix for that matter).

 Matrix of the output-series of splits of the general series U(a,b)F and U(a,b,c)F Primus based Secundus based Tertius based Quartus based U(a,b)F U(-Fa+2b, Fb-2a)F U(a,b)F U[a+b, Fb-(a-b)]F U[-(a-b), Fb-(a+b)]F U(a,b,c)F U(-Fa+2b-c, Fb-2a+c)F U[a+ c/(F-2), b+ c/(F-2)]F U[a+b+ 2c/(F-2), Fb-(a-b)+Fc/(F-2)]F U[-(a-b), Fb-(a+b)+c]F

Note how the secundus only acts as 'unity operator' if c=0.