  The purpose of this note is to provide yet another example of the link
between certain conformal geometries and ordinary differential equations, along
the lines of the examples discussed by Nurowski in math.DG/0406400.
  In this particular case, I consider the equivalence problem for 3-plane
fields D on 6-manifolds M that satisfy the nondegeneracy condition that
D+[D,D]=TM I give a solution of the equivalence problem for such D (as Tanaka
has previously), showing that it defines a so(4,3)-valued Cartan connection on
a principal right H-bundle over M where H is the subgroup of SO(4,3) that
stabilizes a null 3-plane in R^{4,3}. Along the way, I observe that there is
associated to each such D a canonical conformal structure of split type on M,
one that depends on two derivatives of the plane field D.
  I show how the primary curvature tensor of the Cartan connection associated
to the equivalence problem for D can be interpreted as the Weyl curvature of
the associated conformal structure and, moreover, show that the split conformal
structures in dimension~6 that arise in this fashion are exactly the ones whose
so(4,4)-valued Cartan connection admits a reduction to a spin(4,3)-connection.
I also discuss how this case has features that are analogous to those of
Nurowski's examples.
