Einstein normal homogeneous Riemannian manifold (Q676774)

Let \(G\) be a compact connected semi-simple Lie group and \(H\) a closed subgroup with \(\mathfrak g\) and \(\mathfrak h\) their corresponding Lie algebras. Let \(B\) be the Killing form of \(\mathfrak g\) and \(g_0\) the normal homogeneous metric on \(G/H\) induced by \(B\). Consider the decomposition \({\mathfrak g}={\mathfrak h}+{\mathfrak m}\) with \({\mathfrak m}={\mathfrak m}_1 +{\mathfrak m}_2\) satisfying \([{\mathfrak m}_1,{\mathfrak m}_1]\subset {\mathfrak h}\) and \([{\mathfrak m}_2,{\mathfrak m}_2]\subset {\mathfrak h} +{\mathfrak m}_1\). The author shows that if \({\mathfrak m}_1\) and \({\mathfrak m}_2\) are inequivalent irreducible spaces, then \((G/H,g_0)\) is Einstein if and only if \(d_2=2d_1\), where \(d_i=\dim_\mathbb{R} {\mathfrak m}_i\).