Maximal compact subgroup of a nilpotent Lie group

Question

Let $G$ be a connected nilpotent Lie group and $K \subset G$ a maximal compact subgroup. Can we prove that $K$ is always normal?

Answer 1

Yes: $K$ is indeed central.

Consider the adjoint representation of $G$ on $\mathfrak{g}$. Since $G$ is nilpotent, it can be made upper unipotent (= upper triangular with 1 on the diagonal) on a suitable basis. It maps $K$ to a compact subgroup of the group $H$ of such upper unipotent matrices. But in $H$, every non-identity element generates a closed non-compact subgroup, so the only compact subgroup of $H$ is $\{1\}$. So $K$ is contained in the kernel of the adjoint representation, so is central.