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If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{\theta_{0}}(a_{0},a_{R}^{n}) = n \cdot V_{\theta_{0}} (a_{0},a_{R})$ for the generalizedVaserstein symbol whenever $R$ is a smooth affine algebra over a perfect field$k$ with $char(k) \neq 2$ such that $-1 \in {k^{\times}}^{2}$. This enables usto generalize a result of Fasel-Rao-Swan on transformations of unimodular rowsvia elementary matrices over normal affine algebras of dimension $d \geq 4$over algebraically closed fields of characteristic $\neq 2$. As a consequence,we prove that any projective module of rank $3$ with trivial Chern classes overa smooth affine algebra of dimension $4$ over an algebraically closed field $k$with $char(k) \neq 2,3$ is cancellative.

Cancellation of vector bundles of rank $3$ with trivial Chern classes on smooth affine fourfolds