Six-dimensional quadratic forms

Let F be a held of characteristic not equal to 2 and phi be an anisotropic quadratic form of dimension 6 and signed discriminant d not equal 1 such th at phi(F)(root d) is anisotropic. Using a generic method, we give a complet e characterization of quadratic forms psi of dimension greater than or equa l to 4 such that phi becomes isotropic over the function field of the proje ctive quadric defined by the equation psi = 0 (if dim psi = 4, we must assu me d+/-psi, is not an element of (1, d+/-phi)). This method also allows us to recover the results of D. W. HOFFMANN [9], [10] in dimension 5, 6. This settles the study of isotropy of 6-dimensional quadratic forms over the fun ction field of a quadric, except for the cases: 1) phi(F)(root d) is isotropic but not hyperbolic and 1 psi has dimension 4 but is not similar to a 2-fold Pfister form. 2) phi(F(root d)) is anisotropic, dim psi = 4 and d+/-phi = d+/-psi.