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In this paper we prove that the compact Lie group $G_2$ admits aleft-invariant Einstein metric that is not geodesic orbit. In order to provethe required assertion, we develop some special tools for geodesic orbitRiemannian manifolds. It should be noted that a suitable metric is discoveredin a recent paper by I. Chrysikos and Y. Sakane, where the authors proved alsothat this metric is not naturally reductive.

On left-invariant Einstein Riemannian metrics that are not geodesic orbit