Theorem: Homogeneity and strict convexity suffice

A homogeneous strictly convex proper cone in dimension $m\ge 3$ is linearly isomorphic to $L_m$.

Proof. Ito and Lourenço classify homogeneous cones with strictly convex closure: up to linear isomorphism the possibilities are $\{0\}$, ${\mathbb{R}}_{\ge 0}$, ${\mathbb{R}}_{\ge 0}^2$, and Lorentz cones ItoLourenco (Theorem 7). Apply their result to the homogeneous cone $K$, which equals its closure. Full dimension $m\ge 3$ excludes the first three possibilities.