Proposition 15

Let $G$ be a finite-dimensional connected nilpotent Lie group over $\mathbf{R}$ or $\mathbf{C}$ and $A$ an integral subgroup of $G$. Then $Z_G(A)$ is the connected Lie subgroup of $G$ with Lie algebra $\mathfrak{z}_G(L(A))$.

By Proposition 9 of no. 3 it suffices to prove that $Z_G(A)$ is connected. Let $g \in Z_G(A)$. There exists $x \in L(G)$ such that $g = \exp x$ (Proposition 14). Then $\mathrm{Ad}\, g|L(A) = 1$ (no. 3, Proposition 9), hence $\mathrm{Ad}\, g^n|L(A) = 1$ for all $n \in \mathbf{Z}$ and hence $\exp(\mathrm{ad}\, nx)|L(A) = 1$ for all $n \in \mathbf{Z}$. As the mapping

of $K$ into $\mathscr{L}(L(A), L(G))$ is polynomial, $\exp(\mathrm{ad}\, \lambda x)|L(A) = 1$ for all $\lambda \in K$, that is $\exp(\lambda x) \in Z_G(A)$ for all $\lambda \in K$.