**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 C_{2} 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 C_{2} in SO(4) is isomorphic to SO(3) × SO(3). The factor groups of C_{2} 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).

Read more about this topic: Rotations In 4-dimensional Euclidean Space, Geometry of 4D Rotations

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