Invariant Lie Algebra Confinement in a Symmetric Double-Bracket Flow

This paper studies a coupled nonlinear matrix system on the space of real symmetric matrices defined by symmetric double-commutator interactions. The principal result is an algebraic confinement theorem: trajectories remain inside the Lie algebra generated by the initial conditions, so no new Lie-algebraic structure is created dynamically during the evolution. Additional structural properties are established, including preservation of symmetry, conservation of Frobenius norms, and global existence of solutions. A genericity result shows that for generic initial conditions the confining algebra is all of sl(n,R), while exceptional cases form a proper algebraic subvariety. A numerical illustration for n = 2 demonstrates the confinement structure and conserved norm behavior. More broadly, the work suggests a general principle for Lie-polynomial dynamical systems: evolution remains constrained to the algebraic envelope generated at initialization. The paper is intended as a foundational structural note connecting nonlinear matrix dynamics, Lie-algebraic invariance, and constrained operator evolution.