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 $g=g(phi,r)$ 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 third 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 $e=e(phi,r)$ condition $sr(r)=1$ is also sufficient for gauss decomposition. in other words, $e=huu^-u$, where $h=h(phi,r)=tcap e$. 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.