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