On the structure of the special linear groups over Laurent polynomial rings

In this note we prove the following result. Let $C$ be a regular ring such that $\mathrm{SK}(C)=0$. Then the groups $SL_r\bigl(C\bigl[[T_1,\ldots,T_m]\bigr] \left[X_1^{\pm1},\ldots,X_n^{\pm 1},Y_1,\ldots,Y_s\right]\bigr)$ are generated by elementary matrices for all integers $r\geq\max(3,\dim C+2)$.