Vector Bundles of Rank 4 and A3=D3

Over a scheme X with 2 invertible, we show that a version of “Pascal’s rule” for vector bundles of rank 4 gives an explicit isomorphism between the moduli functors represented by projective homogeneous bundles for reductive group schemes of type A3 and D3. We exploit this to prove that a vector bundle $\mathscr V$ of rank 4 has a sub- or quotient line bundle if and only if the canonical symmetric bilinear form on ${\bigwedge} ^2\mathscr V$ has a lagrangian subspace. Under additional hypotheses on X, we prove that this is equivalent to the vanishing of the Witt-theoretic Euler class $e(\mathscr V)\in W^4(X, {\mathrm det}\ \mathscr V^{\vee})$.