Net subgroups of Chevalley groups. II. Gauss decomposition

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup $\Gamma(\sigma)$ of the Chevalley group $G(\Phi, R)$ of type $\Phi$ over a commutative ring $R$ that corresponds to a net $\sigma$, i.e., to a set $\sigma=(\sigma_\alpha)$, $\alpha\in\Phi$, of ideals $\sigma_\alpha$ of $R$ such that $\sigma_\alpha\sigma_\beta\subseteq\sigma_{\alpha+\beta}$ whenever $\alpha,\beta,\alpha+\beta\in\Phi$. It is proved that if the ring $R$ is semilocal, then $\Gamma(\sigma)$ coincides with the group $\Gamma_0(\sigma)$ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of $\Gamma(\sigma)$ into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.