The classification of compact homogeneous spaces of the form \(M=G/K\), where G is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are 4 infinite families and 3 isolated spaces found by Nikonorov and Rodionov in the 90 s. In this paper, we prove that most of these standard Einstein metrics are unstable as critical points of the scalar curvature functional on the manifold of all unit volume G-invariant metrics on M, providing a lower bound for the coindex in the case of Ledger–Obata spaces. On the other hand, examples of stable (in particular, local maxima) invariant Einstein metrics on certain homogeneous spaces of non-simple Lie groups are also given.

\(M=G/K\)形式的紧齐次空间的分类仍然是开放的，其中G是非简单李群，因此标准度量是爱因斯坦。唯一已知的例子是 Nikonorov 和 Rodionov 在 90 年代发现的 4 个无限家庭和 3 个孤立的空间。在本文中，我们证明大多数标准爱因斯坦度量都是不稳定的，因为M上所有单位体积G不变度量的流形上的标量曲率函数的临界点，为 Ledger 的情况下的 coindex 提供了下界 -小畑空间。另一方面，还给出了非简单李群的某些齐次空间上的稳定（特别是局部最大值）不变爱因斯坦度量的示例。