Orthogonal pairs of Euler elements II: Geometric Bisognano--Wichmann and Spin--Statistics Theorems

Models in Algebraic Quantum Field Theory (AQFT) may be generalized including Lie groups of symmetries whose Lie algebras admit an Euler element $h$, characterized by the property that $ad h$ is diagonalizable with eigenvalues in $\{-1, 0, 1\}$. These elements becomes fundamental to the formal description of wedge localization. In this paper, we extend the geometric analysis of Euler wedges and investigate their applications within the AQFT framework. We call a pair of Euler elements $(h, k)$ orthogonal if $e^{i \pi \operatorname{ad} h}(k) = -k.$ Using the geometric framework established in our previous work, we derive both a Bisognano--Wichmann Theorem and a Spin--Statistics Theorem for nets of standard subspaces and von Neumann algebras. Our results {show} how this generalized approach recovers classical results in the AQFT literature while providing a deeper structural understanding of the underlying geometry in established models.