  Let $(\pi, \mathcal{H})$ be a strongly continuous unitary representation of a
1-connected Lie group $G$ such that the Lie algebra $\mathfrak{g}$ of $G$ is
generated by the positive cone $C_\pi := \{x \in \mathfrak{g} : -i\partial
\pi(x) \geq 0\}$ and an element $h$ for which the adjoint representation of $h$
induces a 3-grading of $\mathfrak{g}$. Moreover, suppose that $(\pi,
\mathcal{H})$ extends to an antiunitary representation of the extended Lie
group $G_\tau := G \rtimes \{\mathbf{1}, \tau_G\}$, where $\tau_G$ is an
involutive automorphism of $G$ with $\mathbf{L}(\tau_G) = e^{i\pi\mathrm{ad}
h}$.
  In a recent work by Neeb and \'Olafsson, a method for constructing nets of
standard subspaces of $\mathcal{H}$ indexed by open regions of $G$ has been
introduced and applied in the case where $G$ is semisimple. In this paper, we
extend this construction to general Lie groups $G$, provided the above
assumptions are satisfied and the center of the ideal $\mathfrak{g}_C = C_\pi -
C_\pi$ of $\mathfrak{g}$ is one-dimensional. The case where the center of
$\mathfrak{g}_C$ has more than one dimension will be discussed in a separate
paper.
