The uniqueness of Lyapunov rank among symmetric cones

The Lyapunov rank of a cone is the dimension of the Lie algebra of its automorphism group. It is invariant under linear isomorphism and in general not unique - two or more non-isomorphic cones can share the same Lyapunov rank. It is therefore not possible in general to identify cones using Lyapunov rank. But suppose we look only among symmetric cones. Are there any that can be uniquely identified (up to isomorphism) by their Lyapunov ranks? We provide a complete answer for irreducible cones and make some progress in the general case.