We establish some stability results for $K_1$ of algebras over algebraically closed fields, whose analogues fail for algebras over the reals. More technically, let $k$ be a perfect $C_1$ field and $A$ a non-singular affine algebra of Krull dimension $d \geq 3$ over $k$. Then we show $SL_r(A)/E_r (A)\rightarrow SK_1(A)$ is an isomorphism for $r \geq d+1$ and we find a homomorphism $SL_d (A)\rightarrow WMS_d (A)$ whose kernel is $SL_{d-1} (A)E_d (A)$, which is thus shown to be a normal subgroup.