Over a scheme with 2 invertible, we show that a vector bundle of rank four
has a sub or quotient line bundle if and only if the canonical symmetric
bilinear form on its exterior square has a lagrangian subspace. For this, we
exploit a version of "Pascal's rule" for vector bundles that provides an
explicit isomorphism between the moduli functors represented by projective
homogeneous bundles for reductive group schemes of type A_3 and D_3. Under
additional hypotheses on the scheme (e.g. proper over a field), we show that
the existence of sub or quotient line bundles of a rank four vector bundle is
equivalent to the vanishing of its Witt-theoretic Euler class.Comment: 16 pages, final version; IMRN 2012 rns14
