Theorem: Conditional classification

Suppose an independently constructed representation $(V,\iota ,K)$ satisfies Definition def:representation, $\dim V=m\ge 3$, and $K$ is homogeneous and cone-strictly-convex. Then $K$ is linearly isomorphic to ${\mathcal{L}}_m$.

Proof. Apply the homogeneous strictly convex cone classification of Ito and Lourenço ItoLourenco. Its hypotheses concern the constructed cone. They do not construct the representation.