Non-naturally reductive Einstein metrics on SO(n)

In this article, we obtain that the compact simple Lie group S O ( 4 k + 3 ) ( k ≥ 8 ) admits at least two new non-naturally reductive A d ( S O ( k + 1 ) × S O ( k + 1 ) × S O ( k + 1 ) × S O ( k ) ) -invariant Einstein metrics, and the compact simple Lie group S O ( 4 k + 6 ) ( k ≥ 11 ) admits at least two new non-naturally reductive A d ( S O ( k + 2 ) × S O ( k + 2 ) × S O ( k + 2 ) × S O ( k ) ) -invariant Einstein metrics. Furthermore, we explore the isometry problem for these Einstein metrics. Finally, we prove that there are at least two families of non-naturally reductive invariant Einstein-Randers metrics on S O ( n ) ( n ≥ 35 ) and S O ( n ) ( n ≥ 50 ) respectively.