Optimal injective stability for the symplectic $K_{1}Sp$ group

If $R$ is a commutative ring, $I$ an ideal of $R$ and $v, w \in Um_{2n}(R, I)$thenweshowthat$v, w$ are in the same orbit of elementary action if and only if they are in the same orbit of elementary symplectic action. We also show that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $d!A=A$, $d\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $I$ an ideal of $A$, then $U{m}_d(A,I)=e_1{Sp}_d(A,I)$. As a consequence it is proved that if $A$ is a non-singular affine algebra of dimension $d$ over an algebraically closed field $k$ such that $(d+1)!A=A$, $d\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $I$ a principal ideal then $Sp_{d-1}(A, I) \cap {ESp}{d+1}(A, I) = {ESp}{d -1}(A, I)$. We give an example to show that the above result does not hold true for an affine algebra over a $C_2$ field and also show by an example that the above stability estimate is optimal.