In the settings of Euclidean Jordan algebras and normal decomposition systems (equivalently, Eaton triples), we consider the problems of optimizing linear and distance functions over spectral sets. To present a unified theory, we introduce a new system called Fan-Theobald-von Neumann system which is a triple (V,W,λ), where V and W are two real inner product spaces and λ : V → W is a norm preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. In this general setting, we study linear/distance optimization problems over sets of the form λ⁻¹(Q) and relate the attainment of the optimum value to a commutativity concept. Particular instances include the classical results of Fan, Theobald, and von Neumann and recent results of Tam (on Eaton triples) and Ramirez, Seeger, and Sossa (on Euclidean Jordan algebras).

As an application, we discuss a commutation principle for variational inequalities over such a system.