Skew-hermitian forms which become hyperbolic over a splitting field

Let H = (a, b)(F) be a division quaternion algebra over a field F of characteristic not 2. Denote by tau the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over (H,tau) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h(K) over K. This gives a map of Witt groups rho :W-1(H,r) --> W(K) induced by rho (h) = h(K). When K is a generic splitting field of H we prove in this note that the map rho is injective. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.