n ≥ 6, SO(n) is almost simple in that the factor group SO(n)/C2 of SO(n) by its centre is a simple group. SO(4) is different: there is no conjugation by any element of SO(4) that transforms left- and right-isoclinic rotations into each other. Reflections transform a left-isoclinic rotation into a right-isoclinic one by conjugation, and vice versa. This implies that under the group O(4) of all isometries with fixed point the distinct subgroups and are conjugate to each other, and so cannot be normal subgroups of O(4). The 5D rotation group SO(5) and all higher rotation groups contain subgroups isomorphic to O(4). Like SO(4), all even-dimensional rotation groups contain isoclinic rotations. But unlike SO(4), in SO(6) and all higher even-dimensional rotation groups any two isoclinic rotations through the same angle are conjugate. The set of all isoclinic rotations is not even a subgroup of SO(2), let alone a normal subgroup. Algebra of 4D rotations SO(4) is commonly identified with the group of orientation-preserving isometric linear mappings of a 4D vector space with inner product over the real numbers onto itself. With respect to an orthonormal basis in such a space SO(4) is represented as the group of real 4th-order orthogonal matrices with determinant +1. Isoclinic decomposition A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation as follows: Let be its matrix with respect to an arbitrary orthonormal basis. Calculate from this the so-called associate matrix has rank one and is of unit Euclidean norm as a 16D vector if and only if is indeed a 4D rotation matrix. In this case there exist real numbers and such that and There are exactly two sets of and such that and . They are each other's opposites. The rotation matrix then equals This formula is due to Van Elfrinkhof (1897). The first factor in this decomposition represents a left-isoclinic rotation, the second factor a right-isoclinic rotation. The factors are determined up to the negative 4th-order identity matrix, i.e. the central inversion. Relation to quaternions A point in 4-dimensional space with Cartesian coordinates may be represented by a quaternion . A left-isoclinic rotation is represented by left-multiplication by a unit quaternion . In matrix-vector language this is Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion , which is in matrix-vector form In the preceding section (#Isoclinic decomposition) it is shown how a