A Bruhat decomposition for subgroups containing the group of diagonal matrices. II

This paper is a continuation of RZhMat 1981, 7A438. Suppose $R$ is a commutative ring generated by its group of units $R^*$ and there exist such that. Suppose also that $\mathfrak J$ is the Jacobson radical of $R$, and $B(\mathfrak J)$ is a subgroup of $GL(n,R)$ consisting of the matrices $a=(a_{ij})$ such that $a_{ij}\in\mathfrak J $ for$i>j$. If a matrix $a\in B(\mathfrak J)$ is represented in the form $a=udv$, where $u$ is upper unitriangular, $d$ is diagonal, and $v$ is lower unitriangular, then $u,v\in\langle D,ada^{-1}\rangle$, where $D=D(n,R)$ is the group of diagonal matrices. In particular, $D$ is abnormal in $B(\mathfrak J)$.