Riemannian and Randers Einstein Metrics on SO(n) Which Are Non-naturally Reductive

In this article, we prove that there are at least two new non-naturally reductive Ad(SO(l) × SO(k) × SO(k) × SO(k))-invariant Einstein metrics on compact simple Lie group SO(l + 3k) (k < l). It implies that every compact simple Lie group SO(n) (n > 12) admits at least \(2\left(\left[{n-1\over{4}}\right]-2\right)\) $$2(\left[\frac{n-1}{4}\right]-2)$$ non-naturally reductive left invariant Einstein metrics. Moreover, we obtain that there are at least two families of invariant Einstein–Randers metrics on the compact Lie group SO(n) (n > 12). Besides, we construct non-naturally reductive left invariant (α, β) metrics on the Lie group SO(n) (n > 12). Finally, we examine the isometric problem for those Riemannian Einstein metrics.