Assumption: Symmetric-cone selection axiom

For a proper cone $K\subset V$, assume:

1. $\mathrm{Aut}(K)=\{A\in \mathrm{GL}(V):AK=K\}$ acts transitively on $\mathrm{int}K$;

2. there is a positive-definite inner product $h$ such that

   $$K=K^{\ast h}:=\{y:h(y,x)\ge 0\text{ for all }x\in K\};$$

3. there is no decomposition $V=V_1\oplus V_2$ into positive-dimensional subspaces with $K=K_1\oplus K_2$, where $K_i$ is proper in $V_i$;

4. $K$ is strictly convex.