Proposition 16

Let $G$ be a finite-dimensional nilpotent Lie group over $\mathbf{R}$ or $\mathbf{C}$ and $A$ an integral subgroup of $G$ distinct from $G$. Then $N_G(A)$ is a connected Lie subgroup of $G$ distinct from $A$.

$N_G(A) \neq A$ (Algebra, Chapter I, § 6, Corollary 1 to Proposition 8). By Proposition 11 of no. 4, we need only prove that $N_G(A)$ is connected. Let $g \in N_G(A)$. There exists $x \in L(G)$ such that $g = \exp x$ (Proposition 14). Let $E$ be the vector subspace of $\mathscr{L}(L(G))$ consisting of the $u \in \mathscr{L}(L(G))$ such that $u(L(A)) \subset L(A)$. Then $\mathrm{Ad}\, g^n \in E$ and hence $\exp(\mathrm{ad}\, nx) \in E$ for all $n \in \mathbf{Z}$. Hence $\exp(\mathrm{ad}\, \lambda x) \in E$ for all $\lambda \in K$, that is $\exp(\lambda x) \in N_G(A)$ for all $\lambda \in K$.