On the non-injectivity of the Vaserstein symbol in dimension three

L.N. Vaserstein proved over a two dimensional ring that the orbit space of unimodular rows of length three modulo elementary action has a Witt group structure. R.A. Rao and W. van der Kallen showed that the Vaserstein symbol need not be injective over three dimensional affine algebras over the real field, but is injective over three dimensional affine algebras over a field of cohomological dimension one whose characteristic ≠2,3. R.G. Swan, R.A. Rao and J. Fasel gave another example of a real affine algebra of dimension three for which the Vaserstein symbol is not injective. We demonstrate an uncountably infinite family of non-isomorphic affine algebras of dimension three over the real field for which the Vaserstein symbol is not injective.