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.4||with three non collinear points u,v,w.|
|I.6||where and is bijective.|
|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|
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.