  A biased graph is a graph with a class of selected circles ("cycles",
"circuits"), called "balanced", such that no theta subgraph contains exactly
two balanced circles. A biased graph has two natural matroids, the frame
matroid and the lift matroid. A classical question in matroid theory is whether
a matroid can be embedded in a projective geometry. There is no known general
answer, but for matroids of biased graphs it is possible to give algebraic
criteria. Zaslavsky has previously given such criteria for embeddability of
biased-graphic matroids in Desarguesian projective spaces; in this paper we
establish criteria for the remaining case, that is, embeddability in an
arbitrary projective plane that is not necessarily Desarguesian.
  The criteria depend on the embeddability of a quasigroup associated to the
graph into the additive or multiplicative loop of a ternary coordinate ring for
the plane. A 3-node biased graph is equivalent to an abstract partial 3-net;
thus, we have a new algebraic criterion for an abstract 3-net to be realized in
a non-Desarguesian projective plane. We work in terms of a special kind of
3-node biased graph called a biased expansion of a triangle. Our results apply
to all finite 3-node biased graphs because, as we prove, every such biased
graph is a subgraph of a finite biased expansion of a triangle. A biased
expansion of a triangle, in turn, is equivalent to an isostrophe class of
quasigroups, which is equivalent to a $3$-net. Much is not known about
embedding a quasigroup into a ternary ring, so we do not say our criteria are
definitive. For instance, it is not even known whether there is a finite
quasigroup that cannot be embedded in any finite ternary ring. If there is,
then there is a finite rank-3 matroid (of the corresponding biased expansion)
that cannot be embedded in any finite projective plane---a presently unsolved
problem.
