  In previous work, we employed a geometric method of Kazarian to prove
Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D.
Here we refine that approach to obtain formulas for more general loci,
including those coming from all isotropic Grassmannians. In these cases, the
formulas recover the remarkable theta- and eta-polynomials of Buch, Kresch,
Tamvakis, and Wilson. The streamlined geometric approch yields simple and
direct proofs, which proceed in parallel for all four classical types.
  In an appendix, we develop some foundational algebra and prove several
Pfaffian identities. Another appendix establishes a basic formula for classes
in quadric bundles.
