isomorphic to SO(2). All these subgroups are mutually conjugate in SO(4). Each pair of completely orthogonal planes through is the pair of invariant planes of a commutative subgroup of SO(4) isomorphic to . 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 of SO(4), which is isomorphic to the multiplicative group of unit quaternions. All right-isoclinic rotations likewise form a subgroup of SO(4) isomorphic to . Both and are maximal subgroups of SO(4). Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product with normal subgroups and ; both of the corresponding factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to . (This is not SO(4) or a subgroup of it, because and are not disjoint: the identity and the central inversion each belong to both and .) Each 4D rotation is in two ways the product of left- and right-isoclinic rotations and . and are together determined up to the central inversion, i.e. when both and are multiplied by the central inversion their product is again. This implies that is the universal covering group of SO(4) — its unique double cover — and that and are normal subgroups of SO(4). The identity rotation and the central inversion form a group of order 2, which is the centre of SO(4) and of both and . 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 3L by C2 and of 3R by C2 are each isomorphic to SO(3). Similarly, the factor groups of SO(4) by 3L and of SO(4) by 3R are each isomorphic to SO(3). The topology of SO(4) is the same as that of the Lie group , namely the space where is the real projective space of dimension 3 and is the 3-sphere. However, it is noteworthy that, as a Lie group, SO(4) is not a direct product of Lie groups, and so it is not isomorphic to . Special property of SO(4) among rotation groups in general The odd-dimensional rotation groups do not contain the central inversion and are simple groups. The even-dimensional rotation groups do contain the central inversion and have the group as their centre. For even