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 S3L of SO(4) which is isomorphic to the multiplicative group S3 of unit quaternions. All right-isoclinic rotations likewise form a subgroup S3R of SO(4) isomorphic to S3. Both S3L and S3R are maximal subgroups of SO(4).
Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product S3L × S3R with normal subgroups S3L and S3R; 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 RL and RR. RL and RR are together determined up to the central inversion, i.e. when both RL and RR are multiplied by the central inversion their product is R again.
This implies that S3L × S3R is the double cover of SO(4) and that S3L and S3R 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 S3L and S3R. 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 S3L and S3R are isomorphic to SO(3). The factor groups of S3L and S3R in SO(4) are isomorphic to SO(3).
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