A (closed) positive cone $C$ in a vector space $V$ is called homogeneous if for for all $a$ and $b$ in the interior of $C$ there exists an order isomorphism $\Phi: V\rightarrow V$ (i.e. a linear bijection such that $\Phi(C)=C$) such that $\Phi(a)=b$.
The Koecher-Vinberg theorem states that any homogeneous cone that is self-dual with respect to an inner product is isomorphic to the positive cone of a Euclidean Jordan algebra.
In a more infinite-dimensional setting we could consider JB-algebra's (or JBW)-algebra's. The cones of these algebra's are still homogeneous, but no longer self-dual with respect to an inner product.
So the question is:
Is there some nice property X such that any homogeneous cone with property X is the positive cone of a JB(W)-algebra?
