A note on relative Vaserstein symbol

In an unpublished work of Fasel-Rao-Swan the notion of the relative Witt group $W_E(R,I)$ is defined. In this article we will give the details of this construction. Then we studied the injectivity of the relative Vaserstein symbol $V_{R,I}: Um_3(R,I)/E_3(R,I)\rightarrow W_E(R,I)$. We established injectivity of this symbol if $R$ is an affine non-singular algebra of dimension $3$ over a perfect $C_1$-field and $I$ is a local complete intersection ideal of $R$. It is believed that for a $3$-dimensional affine algebra non-singularity is not necessary for establishing injectivity of the Vaserstein symbol . At the end of the article we will give an example of a singular $3$-dimensional algebra over a perfect $C_1$-field for which the Vaserstein symbol is injective.