A remark on local-global principles for conjugacy classes

A local-global principle is shown to hold for all conjugacy classes of any inner form of GL(n), SL(n), U(n), SU(n), and for all semisimple conjugacy classes in any inner form of Sp(n), over fields k with vcd(k)≤1. Over number fields such a principle is known to hold for any inner form of GL(n) and U(n), and for the split forms of Sp(n), O(n), as well as for SL(p) but not for SL(n), n non-prime. The principle holds for all conjugacy classes in any inner form of GL(n), but not even for semisimple conjugacy classes in Sp(n), over fields k with vcd(k)≤2. The principle for conjugacy classes is related to that for centralizers.