Remark

Let k be a ring, n an integer $> 0$ and A the set of lower triangular matrices with n rows and n columns and elements in k. For $p \geq 0$, let $A_p$ be the set of $(x_{ij}) \in A$ such that $x_{ij} = 0$ for $i - j < p$. Then $A_0 = A$ and $A_pA_q \subset A_{p+q}$. Let $\Gamma_p = 1 + A_p$. Then $\Gamma_1$ is a subgroup of $\mathbf{GL}(n, k)$ called the strict lower triangular group of order n over k. By Proposition 2 of no. 5, $(\Gamma_p)$ is an integral filtration on $\Gamma_1$. As $\Gamma_n = \{1\}$, we see that the group $\Gamma_1$ is nilpotent (Algebra, Chapter I, § 6, no. 3, Definition 6).