I don’t really see your point, sorry? In arbitrary dimension, you rotate in an oriented plane (aka a bivector up to a positive constant) around a fixed point, i

I don’t really see your point, sorry? In arbitrary dimension, you rotate in an oriented plane (aka a bivector up to a positive constant) around a fixed point, if anything; I meant to call that an “axis” by abusing the terminology a bit (it’s an oriented line in three dimensions and a point plus an orientation, i.e. either “clockwise” or “counterclockwise”, in two). Not all isometries with a fixed point will be “rotations” in that sense, though—some will be products of them, as in / cos u sin u 0 0 \ | -sin u cos u 0 0 | | 0 0 cos v sin v | \ 0 0 -sin v cos v / with non-boring values of u and v.