Characterization of minimal Pfister neighbors via Rost projectors

Let φ be an anisotropic nine-dimensional quadratic form over a field (of characteristic ≠2). We show that the projective quadric given by φ possesses a Rost projector if and only if φ is a Pfister neighbor. The following consequence of this result gives the initial step in the construction [10] of the field with the u-invariant 9: if φ is not a Pfister neighbor and the Schur index of its even Clifford algebra is at least 4, then φ stays anisotropic over the function field of any 9-dimensional form nonsimilar to φ.