Remarks on commuting involutions

In [3, p. 293] R. Hermann poses the following problem (without the restriction that G be simple). (A) Given s1 and s2 nontrivial involutive automorphisms of a compact simple Lie group G, find xEG such that Ad(x)s,Ad(x)-l commutes with s2. We wish to discuss the existence of solutions for (A). Without real loss of generality we assume G simply connected. The respective fixed point groups of s1 and s2 are closed connected subgroups K1 and K2 of G, and K1 acts from the left on G/K2. In both [1 I and [3 ] it is shown that there is a flat geodesically imbedded torus TCG/K2 which meets orthogonally every K1-orbit. Furthermore, if the decompositions of the Lie algebra g of G into + 1 and -1 eigenspaces are given respectively by