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

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