On the theory of Pfaffians based on exponential maps in exterior algebras

Based on the relation of exponential maps and interior products in exterior algebras, some formulas of Pfaffians, including expansion formulas and the Cayley–Jacobi formula for determinants of alternating matrices, are deduced with new proofs. As an application, Pfaffian powers of alternating bilinear forms [O. Loos, Discriminant algebras and adjoints of quadratic forms, Beiträge Algebra Geom. 38 (1997) 33–72] are interpreted in terms of exponential maps in algebras of alternating multi-linear forms.