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We obtain new invariant Einstein metrics on the compact Lie groups  $SO(n)$ ($n \geq 7$) which are not naturally reductive. This is achieved byimposing certain symmetry assumptions in the set of all  left-invariant metrics on $SO(n)$ and by computing the Ricci tensor for suchmetrics. The Einstein metrics are obtained as solutions of systems polynomialequations, which we manipulate by symbolic computations using Gr\"obner bases.

New Einstein metrics on the Lie group $SO(n)$ which are not naturally reductive