Question: From every [projective plane](http://en.wikipedia.org/wiki/Projective_plane) a coordinitisation can be constructed on a [planar ternary ring](http://en.wikipedia.org/wiki/Planar_ternary_ring), and conversely from every planar ternary ring a projective plane can be constructed. (For background see [Weibel's survey of non-Desarguian planes](http://www.ams.org/notices/200710/tx071001294p.pdf)).

Isomorphic planar ternary rings yield isomorphic projective planes, however there exist projective planes that can be coordinitised by non-isomorphic planar ternary rings.

Which planar ternary rings coordinitise their projective plane uniquely up to isomorphism? 

(In other words there is a surjective function from isomorphism classes of planar ternary rings to isomorpism classes of projective planes; on what domain is it injective?)

As stated in Weibel's survey above two ternary rings are isomorphic if (and only if) the automorphism group of the projective plane maps any quadrilateral into any other quadrilateral. It is not clear to me, however, what this means about the ternary ring itself.

I know this class includes the alternative division rings (see [Bruck and Kleinfeld - The Structure of Alternative Division Rings, Theorem B in Section 5](http://www.ams.org/journals/proc/1951-002-06/S0002-9939-1951-0045099-9/S0002-9939-1951-0045099-9.pdf)).

P.S. I originally stated that this class contains the near-fields, but this is false.

Answer: 
The answer to the problem is well known; since :

all the PTRs coordinatizing a given projective plane are isomorphic iff it is a moufang plane iff all these PTRS are isomorphic alternative division rings. <http://www.math.uni-kiel.de/geometrie/klein/math/geometry/moufang.html> 

However the problem to give a purely algebraic proof of this result seem to be an open problem.
A necessary and sufficient condition so that two planar ternary rings coordinatize isomorphic projective planes was obtained in 2004 by A.Grari

* *A necessary and sufficient condition so that two planar ternary rings induce isomorphic projective planes*. Arch. Math. **83**, (2004) 183–192, doi:[10.1007/s00013-003-4580-9](https://doi.org/10.1007/s00013-003-4580-9).

The well-known theorems on skew-fields and division rings and alternative division ring can be deduced from this condition as particular cases.

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Response added later:

Let $\pi$ be a projective plane coordinatized by a unique (up to isomorphism) Hall ternary ring $S$. If $\pi$ is finite then by the theorem of Ostrom-Wagner
(see eg Kantor's 1985 *[Flag-Transitive Planes](http://pages.uoregon.edu/kantor/PAPERS/FlagTraPlanes.pdf)*)
$\pi$ is desarguesian; hence $S$ is a skew-field, and therefore there is no problem to solve in this case since the solution is well known.

If $\pi$ is infinite then $\pi$ is a flag-transitive plane so all the PTRs coordinatizing this plane are equivalent under the operation of comparison (see lemma 4.1 of Grari's paper above).

The problem posed can now be reduced to the following:
When are two equivalent (under comparison) Hall ternary rings isomorphic?