Generalized conference matrices and projective planes

In 1982, Dieter Jungnickel showed that the existence of an ( a , A ) -transitive finite projective plane with a nonincident point-line pair ( a , A ) is equivalent to the existence of a generalized conference matrix of index 1 over the group of all ( a , A ) -homologies. I will further investigate this equivalence with respect to projective planes of Lenz–Barlotti classes I.2, I.3, I.4, and II.2. As an application, I will obtain a short proof of the following result: if a finite projective plane of even order is ( a , A ) - and ( b , B ) -transitive with B = a b and b = A B and if the group of all ( a , A ) -homologies is abelian, then the group of all ( b , B ) -elations is elementary abelian and therefore the order of the plane is a power of 2.