A metric $g$ on a simply connected manifold $M$ is called the second best Einstein metric, if $(M,g)$ is an Einstein manifold with positive scalar curvature which is non isometric to the sphere and its curvature operator $R$ minimizes the angle $\sphericalangle(R(p), Id)$ to the identity at each point $p \in M$ among all Einstein manifolds with the properties above. In this talk we use the identity $2(\text{scal}/n) R = \Delta R + 2(R^2+R^{\#})$ that holds for all Einstein manifolds in order to present an approach towards finding the second best Einstein metric in dimensions $\leq 11$.