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**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 | Primus doubled | Tertius*(F+2) doubled | Secundus doubled | Quartus doubled | Tertius doubled | Primus*(F+2) doubled | Quartus doubled | Secundus doubled | |||||||||||||||||

U_{-5} | -F^{5}-F^{4}+4F^{3}+3F^{2}-3F-1 | = | -F | * | (F^{3}-F^{2}-2F+1)*(F+2) | + | -F-1 | = | F^{2}-2 | * | -F^{3}-F^{2}+2F+1 | + | F+1 | = | F^{2}-F-1 | * | -(F^{2}-1)*(F+2) | + | 1 | = | -F^{2}-F+1 | * | F^{3}-3F | + | -1 |

U_{-4} | -F^{4}-F^{3}+3F^{2}+2F-1 | = | -F | * | (F^{2}-F-1)*(F+2) | + | -1 | = | F^{2}-2 | * | -F^{2}-F+1 | + | 1 | = | F^{2}-F-1 | * | -F*(F+2) | + | -1 | = | -F^{2}-F+1 | * | F^{2}-2 | + | 1 |

U_{-3} | -F^{3}-F^{2}+2F+1 | = | -1 | * | (F^{2}-F-1)*(F+2) | + | -F-1 | = | F | * | -F^{2}-F+1 | + | F+1 | = | F-1 | * | -F*(F+2) | + | 1 | = | -F-1 | * | F^{2}-2 | + | -1 |

U_{-2} | -F^{2}-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 |

U_{0} | -1 | = | 0 | * | F+2 | + | -1 | = | 2 | * | -1 | + | 1 | = | 1 | * | 0 | + | -1 | = | -1 | * | 2 | + | 1 |

U_{1} | 1 | = | 1 | * | F+2 | + | -F-1 | = | F | * | -1 | + | F+1 | = | 1 | * | 0 | + | 1 | = | 1 | * | 2 | + | -1 |

U_{2} | F+1 | = | 1 | * | F+2 | + | -1 | = | F | * | 1 | + | 1 | = | 1 | * | F+2 | + | -1 | = | 1 | * | F | + | 1 |

U_{3} | F^{2}+F-1 | = | F | * | F+2 | + | -F-1 | = | F^{2}-2 | * | 1 | + | F+1 | = | F-1 | * | F+2 | + | 1 | = | F+1 | * | F | + | -1 |

U_{4} | F^{3}+F^{2}-2F-1 | = | F | * | (F-1)*(F+2) | + | -1 | = | F^{2}-2 | * | F+1 | + | 1 | = | F-1 | * | F*(F+2) | + | -1 | = | F+1 | * | F^{2}-2 | + | 1 |

U_{5} | F^{4}+F^{3}-3F^{2}-2F+1 | = | F^{2}-1 | * | (F-1)*(F+2) | + | -F-1 | = | F^{3}-3F | * | F+1 | + | F+1 | = | F^{2}-F-1 | * | F*(F+2) | + | 1 | = | F^{2}+F-1 | * | F^{2}-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.