Did you make a typing error here? In the second paragraph, did you really mean C, not B? If so, that is correctHere AB is 'timelike' , so that there is a frame of reference in which A & B occur in same location, but just separated in time, and if in that frame A precedes B, than A precedes B in all frame(s).
But this is not true for spacelike separated events as are A & C, there is a frame in which A and B occur simultaneously separated only in space, but there are also frames in which A precedes B (as the given) and frames in which B precedes A,...
but that isn't. "Causality" refers only to the ordering of timelike-separated events (like A and B) or null-separated events....so that causality may not be maintained.