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 12) admits at least 2 (n-14-2) 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.