Gauss decomposition for Chevalley groups, revisited

In the 1960’s Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that Chevalley groups $G=G(\mathrm{\Phi },R)$ over a semilocal ring admit remarkable Gauss decomposition $G=TU{U}^{-}U$, where $T=T(\mathrm{\Phi },R)$ is a split maximal torus, whereas $U=U(\mathrm{\Phi },R)$ and $U^{-}=U^{-}(\mathrm{\Phi },R)$ are unipotent radicals of two opposite Borel subgroups $B=B(\mathrm{\Phi },R)$ and $B^{-}=B^{-}(\mathrm{\Phi },R)$ containing $T$. It follows from the classical work of Hyman Bass and Michael Stein that for classical groups Gauss decomposition holds under weaker assumptions such as $\text{sr}(R)=1$ or $\text{asr}(R)=1$. Later the second author noticed that condition $\text{sr}(R)=1$ is necessary for Gauss decomposition. Here, we show that a slight variation of Tavgen’s rank reduction theorem implies that for the elementary group $E(\mathrm{\Phi },R)$ condition $\text{sr}(R)=1$ is also sufficient for Gauss decomposition. In other words, $E=HU{U}^{-}U$, where $H=H(\mathrm{\Phi },R)=T\cap E$. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, $\text{asr}(R)=1$, $\text{sr}(R,\mathrm{\Lambda })=1$, etc., were only needed to guarantee that for simply connected groups $G=E$, rather than to verify the Gauss decomposition itself.