Proposition 36

*Let I be a subset of $\{1, 2, \ldots, n\}$ and G the subgroup of $\mathbf{GL}(E)$ consisting of the $g = (g_{ij})_{1 \leq i, j \leq n} \in \mathbf{GL}(E)$ such that $g_{ij} = 0$ for $i < j$ and $g_{ii} = 1$ for $i \in I$. Then G is a Lie subgroup of $\mathbf{GL}(E)$ and $\mathbf{L}(G)$ is the set of $x = (x_{ij})_{1 \leq i, j \leq n} \in \mathscr{L}(E)$ such that $x_{ij} = 0$ for $i < j$ and $x_{ii} = 0$ for $i \in I$.

Let S be the set of $(x_{ij}) \in \mathscr{L}(E)$ such that $x_{ij} = 0$ for $i < j$ and $x_{ii} = 0$ for $i \in I$. Then G is the intersection of $\mathbf{GL}(E)$ and the affine subspace $1 + S$ of $\mathscr{L}(E)$. Hence G is a submanifold of $\mathbf{GL}(E)$ and the tangent space to G at 1 is identified with S.*

In particular, in $\mathbf{GL}(n, K)$, the total lower triangular subgroup and the lower strict triangular subgroup, defined as in Integration, Chapter VII, § 3, no. 3, are Lie subgroups with Lie algebras $t(n, K)$ and $n(n, K)$ (Chapter I, § 1, no. 2).