Obviously, the Lenz-Barlotti figure is invariant under the action of , and

is the Lenz figure of . The Lenz-Barlotti classification lists all possible structure types for the Lenz-Barlotti figure of a projective plane. Each of the Lenz types will be divided in one or more Lenz-Barlotti types.

**Theorem.** Let
be a projective plane.
Then exactly one of the following statements is true.

Type | Lenz-Barlotti figure | Coordinatizing ternary field |
---|---|---|

I.1 | Ternary fields | |

I.2 | ||

I.3 | ||

I.4 | with three non collinear points u,v,w. | |

I.6 | where and is bijective. | |

II.1 | Cartesian groups | |

II.2 | ||

III.1 | where . | |

III.2 | where . | |

IVa.1 | for a line a. | Quasifields |

IVa.2 | There exist a line a und two distinct point u,v on a such that | Nearfields |

IVa.3 | There exist a line a and an involutorial, fixed point free, permutation t of a such that | |

IVb.1 | Dual of IVa.1 | |

IVb.2 | Dual of IVa.2 | |

IVb.3 | Dual of IVa.3 | |

V | There exist a line a and a point z on a such that | Semifields |

VII.1 | Alternative fields | |

VII.2 | Skewfields |

The original list of all Lenz-Barlotti types was slightly longer,
but, as it turned out, some types does not exist. These
missing types are
**I.5, I.7, I.8** and **II.3**. Whether there exists
a projective plane of
Lenz-Barlotti type **I.6** seems to be an open problem.
The Lenz-Barlotti type **IVa.3**, and its dual **IVb.3**,
is somewhat exceptional. In fact, there is
exactly one plane of this type.

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