Realizations of subgroups of type \(D_8\) of connected exceptional simple Lie groups of type \(E_8\) (Q5954678)

I. Yokota and his colleagues have determined all involutive automorphisms \({\sigma}\) of an exceptional simple Lie group \(G\) and realized the subgroups \(G^{\sigma}\) of fixed points of \({\sigma}\) [cf. e.g., \textit{I. Yokota}, Tsukuba J. Math. 14, 185-223, 379-404 (1990; Zbl 0732.22002, Zbl 0741.22004); 15, 301-314 (1991; Zbl 0761.22019) and \textit{I. Yokota} and \textit{O. Yasukura}, Tsukuba J. Math. 10, 331-349 (1986; Zbl 0618.22009)] which correspond to Berger's results for simple Lie algebras [\textit{M. Berger}, Ann. Sci. Éc. Norm. Supér. (3) 74, 85-177 (1959; Zbl 0093.35602)]. In the article under review, the author realizes subgroups of type \(D_8\) of the compact and non-compact Lie groups of type \(E_8\). The realizations are explicit, using substantially the spinor groups. Compared to the previous works mentioned above, for the type \(D_8\) subgroups of \(E_8\), these realizations are simpler and more intelligible, and should be more applicable to the study of symmetric spaces.