Let $R$ be a commutative ring and denote by ${\rm GL}_n(R)$ the general linear group over $R$. Let $I_n$ denote the identity matrix of size $n$. Let $G$ be a subgroup of ${\rm GL}_n(R)$ such that every element(matrix) $A$ in $G$ satisfies the property that every entry of the matrix $A-I_{n}$ is nilpotent, i.e. if we write $A-I_n=(a_{ij})$, then there exists a positive integer $r$ such that $a_{ij}^{r}=0$ for all $i,j$.
Question: Is the group $G$ solvable?
My understanding is the following: Under the assumption, every element $g$ of $G$ should be unipotent, i.e. $(g-1)^m=0$ for some integer $m$. From this, one should have the lower central series of $G$ terminates with $\{1\}$. Thus, $G$ is nilpotent, and hence it is solvable.