Related pagesThe LenzBarlotti classification 
Obviously, the Lenz figure is invariant under the action of . The Lenz classification of projective planes states that there are only 7 possible types of Lenz figures.
Theorem. Let be a projective plane. Then exactly one of the following seven statements is true.
Lenz type  Lenz figure  Coordinatizing ternary field 

I  Ternary fields  
II  Cartesian groups  
III  There exist a point z and a line l with
such that

Special Cartesian groups 
IVa  There exists a line a such that  Quasifields 
IVb  There exists a point z such that  Dual of IVa 
V  There exist a line a and a point z on a such that  Semifields 
VII  Alternative fields 
The Lenz types IVa,V and VII are translation planes. In terms of translation axes, we get the following table.
Lenz type  Interpretation 

IVa  Translation planes with a unique translation axis. 
IVb  Dual translation planes with unique translation center 
V  The plane is a translation plane and a dual translation plane. 
VII  Moufang planes 
In particular, we have the theorem of Skornjakov and SanSoucie.
Theorem. A projective plane which admits two distinct translation axes is a Moufang plane.