Fischer spaces and Lie algebras

A Fischer space is a partial linear space (P,L) in which
each line contains 3 points and two intersecting lines
generate a subspace isomorphic to an affine or dual affine plane.

Starting from a Fischer space (P,L) one can construct an
algebra on the vector space 2^P defined by the rule that

                 p*q=p+q+r if {p,q,r} is a line
                    =0     if p and q are not collinear or p=q.

We study the structure of this algebra and determine the
simple Lie algebras that arise from this construction.

As a consequence of our work we find a new proof of the fact that
the group F_22 is a subgroup of the group ^2 E_6(2).