2-tori in \(E_ 8\) (Q1091486)

This note determines the conjugacy classes of maximal elementary abelian 2-subgroups in the compact exceptional Lie group \(E_ 8\). There are two classes, one of rank 9 and one of rank 8. The proof depends on the corresponding result for the subgroups \(Ss^+(16)\subset E_ 8\) (four classes). This in turn is deduced from results given for Spin(16) (two classe), PSO(n) (where \(n\equiv 0\) mod 8 and \(n>8)\), and PO(n) (where \(n\equiv 0\) mod 2).