  Involutivity is the algebraic property that guarantees solutions to an
analytic and torsion-free exterior differential system or partial differential
equation via the Cartan-K\"ahler theorem. Guillemin normal form establishes
that the prolonged symbol of an involutive system admits a commutativity
property on certain subspaces of the prolonged tableau. This article examines
Guillemin normal form in detail, aiming at a more systematic approach to
classifying involutive systems. The main result is an explicit quadratic
condition for involutivity of the type suggested but not completed in Chapter
IV, \S 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner,
Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and
characterizes involutive tableaux.
