  Using the Rost invariant for torsors under Spin groups one may define an
analogue of the Arason invariant for certain hermitian forms and orthogonal
involutions. We calculate this invariant explicitly in various cases, and use
it to associate to every orthogonal involution with trivial discriminant and
trivial Clifford invariant over a central simple algebra of even co-index a
cohomology class $f_3$ of degree 3 with $\mu_2$ coefficients. This invariant
$f_3$ is the double of any representative of the Arason invariant; it vanishes
when the algebra has degree at most 10, and also when there is a quadratic
extension of the center that simultaneously splits the algebra and makes the
involution hyperbolic. The paper provides a detailed study of both invariants,
with particular attention to the degree 12 case, and to the relation with the
existence of a quadratic splitting field.
