Jordan and isometric cone automorphisms in Euclidean Jordan algebras

Abstract

Every symmetric cone $K$ arises as the cone of squares in a
Euclidean Jordan algebra $V$. As $V$ is a real inner-product
space, we may denote by $\mathrm{Isom}(V)$ its group of isometries. The
groups $\mathrm{JAut}(V)$ of its Jordan-algebra automorphisms and
$\mathrm{Aut}(K)$ of the linear cone automorphisms are then related. For
certain 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.