Symplectic completion over smooth affine algebras

In this article, we prove the following results:\ \noindent $\text{(1).}$ Let $R$ be a smooth affine algebra of dimension $3$ over an algebraically closed field $K$ with $3!\in K$, then we show that ${\text{Um}}_4(R)=e_1{\text{Sp}}_4(R)$ and ${\text{Um}}_4(R[X])=e_1{\text{Sp}}_4(R[X])$. \noindent $\text{(2).}$ We also show that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that ${\text{W}}_E(R)$ is divisible, then ${\text{Um}}_3(R)=e_1{\text{SL}}_3(R)$. As a consequence it is shown that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that ${\text{W}}_E(R)$ is divisible, then ${\text{Um}}_4(R)=e_1{\text{Sp}}_4(R)$. \noindent $\text{(3).}$ We show that if $R$ is a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then ${\text{Um}}_4(R[X])=e_1{\text{Sp}}_4(R[X])$. \noindent $\text{(4).}$ We also show that if $R={\oplus }_{i\ge 0}{R}_i$ is a graded ring over a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then ${\text{Um}}_4(R)=e_1{\text{Sp}}_4(R)$.