A single induction proof of Simons' Riemannian holonomy theorem

We give a detailed and complete algebraic proof of Simons’ Riemannian holonomy theorem by a single strong induction on the dimension of the linear span of the curvature orbit. The proof uses the construction of flats and root centralizers, but replaces the detailed analysis of the common zero-weight space, the most technical core of previous algebraic proofs, by a trace-vanishing lemma. The lemma applies to curvature operators contained in an ideal that annihilates the curvature module. Restriction to a common geodesic subspace produces a proper invariant kernel. Induction makes this kernel fixed by the holonomy action, after which the trace lemma shows that the root centralizers span the ambient space. Their intersection recovers the maximal flat, and irreducibility completes the proof. For the completeness we also include the detailed derivation of the local symmetry of the underlying Riemannian manifold from the algebraic theorem.