  In a previous article, a universal linear algebraic model was proposed for
describing homogeneous conformal geometries, such as the spherical, Euclidean,
hyperbolic, Minkowski, anti-de Sitter and Galilei planes. This formalism was
independent from the underlying field, providing an extension and general
approach to other fields, such as finite fields. Some steps were taken even for
the characteristic $2$ case.
  In this article, we undertake the study of the characteristic $2$ case in
more detail. In particular, the concept of virtual quadratic spaces is used,
defined in a previous article by the author, and a similar result is achieved
for finite fields of characteristic $2$ as for other fields. Some differences
from the non-characteristic $2$ case are also pointed out.
