Jordan and isometric cone automorphisms in Euclidean Jordan algebras

Every symmetric cone $K$ arises as the cone of squares in aEuclidean Jordan algebra $V$. As $V$ is a real inner-productspace, we may denote by $\mathrm{Isom}(V)$ its group of isometries. Thegroups $\mathrm{JAut}(V)$ of its Jordan-algebra automorphisms and$\mathrm{Aut}(K)$ of the linear cone automorphisms are then related. Forcertain inner products,\begin{equation*}\mathrm{JAut}(V)=\mathrm{Aut}(K) \cap \mathrm{Isom}(V).\end{equation*}We characterize the inner products for which this holds.