Looking at generators, the double symmetry is seen as adding a new operator that maps symmetric positions in the Coxeter diagram, making some original generators redundant. For 3D space groups, and 4D point groups, Coxeter defines an index two subgroup of [[X]], [[X] ], which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to [[X]], which is the chiral subgroup of [[X]]. So for example the 3D space groups [[4,3,4]] (I432, 211) and [[4,3,4] ] (Pm 3 n, 223) are distinct subgroups of [[4,3,4]] (Im 3 m, 229).