Rotations in 4-dimensional Euclidean Space - Geometry of 4D Rotations - Group Structure of SO(4)

Group Structure of SO(4)

SO(4) is a noncommutative compact 6-parameter Lie group.

Each plane through the rotation centre O is the axis-plane of a commutative subgroup isomorphic to SO(2). All these subgroups are mutually conjugate in SO(4).

Each pair of completely orthogonal planes through O is the pair of invariant planes of a commutative subgroup of SO(4) isomorphic to SO(2) × SO(2).

These groups are maximal tori of SO(4), which are all mutually conjugate in SO(4). See also Clifford torus.

All left-isoclinic rotations form a noncommutative subgroup S3_L of SO(4) which is isomorphic to the multiplicative group S3 of unit quaternions. All right-isoclinic rotations likewise form a subgroup S3_R of SO(4) isomorphic to S3. Both S3_L and S3_R are maximal subgroups of SO(4).

Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product S3_L × S3_R with normal subgroups S3_L and S3_R; both of the corresponding factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to S3.

Each 4D rotation R is in two ways the product of left- and right-isoclinic rotations R_L and R_R. R_L and R_R are together determined up to the central inversion, i.e. when both R_L and R_R are multiplied by the central inversion their product is R again.

This implies that S3_L × S3_R is the double cover of SO(4) and that S3_L and S3_R are normal subgroups of SO(4). The non-rotation I and the central inversion -I form a group C₂ of order 2, which is the centre of SO(4) and of both S3_L and S3_R. The centre of a group is a normal subgroup of that group. The factor group of C₂ in SO(4) is isomorphic to SO(3) × SO(3). The factor groups of C₂ in S3_L and S3_R are isomorphic to SO(3). The factor groups of S3_L and S3_R in SO(4) are isomorphic to SO(3).