  Let Gr(2, E) be the Grassmann bundle of two-planes associated to a general
bundle E over a curve X. We prove that an embedding of Gr(2, E) by a certain
twist of the relative Pl\"ucker map is not secant defective. This yields a new
and more geometric proof of the Hirschowitz-type bound on the Lagrangian Segre
invariant for orthogonal bundles over X, analogous to those given for vector
bundles and symplectic bundles in [2, 3]. From the non-defectivity we also
deduce an interesting feature of a general orthogonal bundle over X,
contrasting with the classical and symplectic cases: Any maximal Lagrangian
subbundle intersects at least one other maximal Lagrangian subbundle in
positive rank.
