We compute the unramified cohomology of quadrics of dimension 4 in degree 4 over an arbitrary field of characteristic different from 2. We find that it is related to classical invariants of a more elementary nature, such as the group of spinor norms and the projective special orthogonal group modulo Manin's R-equivalence.
"Unramified cohomology of quadrics, II." Duke Math. J. 106 (3) 449 - 484, 15 February 2001. https://doi.org/10.1215/S0012-7094-01-10632-7