Characterization of metric 4-forms

Characterize the precise condition under which a 4-form on an 8-dimensional real vector space is metric, meaning that the Karigiannis quantity q_α is proportional to the square of a non-degenerate quadratic form.

Background

The paper defines a 4-form α on an 8-dimensional real vector space to be metric when q_α is proportional to the square of a non-degenerate quadratic form. Cayley and split Cayley forms provide known examples, and the paper constructs additional SU(4)-stabilized examples that define non-degenerate Riemannian metrics through the same formula.

Although the examples suggest a relationship between metricity and stabilization by subgroups of an orthogonal group, the paper does not establish a necessary and sufficient condition. It also notes that compactness of the stabilizer may not be required except when restricting to Riemannian signature.