Correction: The geometry of compact homogeneous spaces with two isotropy summands

We give a complete list of all homogeneous spaces \(M=G/H\) $M=G/H$ where G is a simple compact Lie group, H a connected, closed subgroup, and G/H is simply connected, for which the isotropy representation of H on \(T_pM\) $T_{p}M$ decomposes into exactly two irreducible summands. For each homogeneous space, we determine whether it admits a G-invariant Einstein metric. When there is an intermediate subgroup \(H< K < G\) $H<K<G$ , we classify all the G-invariant Einstein metrics. This is an extension of the classification of isotropy irreducible spaces, given independently by Manturov (Dokl Akad Nauk SSSR 141:792–795, 1961; Dokl Akad Nauk SSSR 141:1034–1037, 1961; Tensor Semin Vector Anal 13:68–145, 1966) and Wolf (Acta Math 120:59–148, 1968; Acta Math 152:141–142, 1984).