Cartan embeddings of compact Riemannian 3-symmetric spaces (Q675758)

Let \(f\) be an automorphism of a compact Lie group \(G\) and \(K=\{x\in G:f(x)=x\}. \) Then the mapping \(F: G\rightarrow G\) defined by \(F(x)=xf(x^{-1})\) induces the ``Cartan embedding'' \(\Psi : G/K \rightarrow G.\) If \(f^{2}=\text{id}\), then the image of this embedding is a totally geodesic submanifold. In his previous work the author has presented the list of involutions \(f\) such that the image of \(\Psi\) is a stable minimal submanifold [\textit{K. Mashimo}, Arch. Math. 58, 500-508 (1992; Zbl 0762.53036)]. The paper under review contains the list of automorphisms \(f\) such that \(f^{3}=\text{id}\) and the image of \(\Psi\) is a minimal submanifold. Moreover, if \(\Psi (G/K)\) is a stable minimal submanifold, then \((G,K)\) is either \((E_{6},(SU(3)\cdot SU(3)\cdot SU(3))\slash Z_{3})\), \((E_{7},(SU(3)\cdot SU(6))\slash Z_{3})\), \((E_{8},(SU(3)\cdot E{_6})\slash Z_{3}),\) \((G_{2},SU(3)),\) \((\text{Spin}(8),G_{2}),\) or \((\text{Spin}(8),SU(3)\slash Z_{3})\).