Let (pi, H) be a strongly continuous unitary representation of a 1-connected Lie group G such that the Lie algebra g of G is generated by the positive cone C-pi := {x is an element of g : -i partial derivative pi(x) >= 0} and an element h for which the adjoint representation of h induces a 3-grading of g. Moreover, suppose that (pi, H) extends to an antiunitary representation of the extended Lie group G(tau) := G (sic) {1, tau(G)}, where tau(G) is an involutive automorphism of G with L(tau(G)) = e(i pi ad h). In a recent work by Neeb and Olafsson, a method for constructing nets of standard subspaces of 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 g(C) = C-pi - C-pi, subset of g is one-dimensional. The case where the center of g(C) has more than one dimension will be discussed in a separate paper.
