Symplectic groups over Laurent polynomial rings and patching diagrams

In this note we prove the following result. Let $A$ be a P.I.D. such that $\operatorname{K}_1\operatorname{Sp}(A)=0$. Then the groups $\operatorname{Sp}_{2r}(A[X_1^{\pm1},\ldots,X_n^{\pm1},Y_1,\ldots,Y_m])$ are generated by elementary symplectic matrices for all integers $r\geq2$.