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.