The Alekseevskii conjecture in 9 and 10 dimensions

We show that noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no Einstein metrics negative Ricci curvature, with only three potential exceptions. The main ingredient in the proof is show, via a cohomogeneity-one approach, admitting an ideal isomorphic $ \mathfrak{sl}_2(\mathbb{R}) curvature.