Odd orthogonal matrices and the non-injectivity of the Vaserstein symbol

R.A. Rao-W. van der Kallen showed that the Vaserstein symbol VΓ(SR3) from the orbit space of unimodular rows of length three over the coordinate ring of the real three sphere SR3 modulo elementary action to the elementary symplectic Witt group WE(Γ(SR3)) is not injective. Dhvanita R. Rao-Neena Gupta gave an uncountable family of singular real threefolds Aα for which the Vaserstein symbol VAα is not injective. In this paper, we give a countable family of smooth real birationally equivalent threefolds An for which the Vaserstein symbol VAn is not injective.