From every projective plane a coordinitisation can be constructed on a 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).

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).

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

Weibel. He wrote a famous book on homological algebra. – Todd Trimble♦ Sep 22 '12 at 23:53