Let $U=U(n,R)$ the group of $n\times n$ matrices over R (a commutative unitary ring) with $1$ on the diagonal and $0$ under it. For each $0<i\leq n$ let $U_i$ be the subgroup of $U$ of all elements of $U$ where the first $i-1$ superdiagonals are zero. This way we have that $1=U_n<\ldots<U_1=U$. It is easily shown by computation that this is a central series of $U$, which is hence a nilpotent group. Is it true that $U/U_i$ is isomorphic with $U(n-i,R)$? I'm not used to work with matrices, so I am not satisfied with any approach I try to enact.
Would you give me a formal, maybe even "nice" description of this isomorphism, if there is one?