I've been reading the Wikipedia article about rotations in four dimensional space, and it has some interesting differences from any other dimension. The group of all rotations of R^n is the "special orthogonal group" denoted as SO(n). It's a Lie group of dimension equal to (n-1) + (n-1) + ... + 1 = n((n-1)/2. SO(1) and SO(2) are respectively the group of order 2 and the circle group. For n ≥ 3, n ≠ 4: ----------------- * If n is odd, n = 3, 5, 7, ...: SO(n) is a simple group*. * If n is even, n = 6, 8, 10, ...: SO(n)/{-I,I} is simple. Here {-I, I} is the group generated by the central inversion -I, and this group of order 2 is the center of SO(n) (the normal subgroup of the elements that commute with the entire group). ----------------- But for SO(4) the situation is unique. Let S^3 denote the unit sphere in 4-space, i.e., the unit quaternions. Then any rotation of R^4 (a map h : R^4 —> R^4 that preserves all distances and orientation) can be described as x —> p x q for some p, q in S^3, where x is any element of R^4 thought of as a quaternion. Given h, the ordered pair (p, q) is almost unique: the only other pair also describing h is (-p, -q). (Because quaternions are associative, no parentheses are needed in x —> p x q.) An "isoclinic" rotation of R^4 is one where all nonzero points rotate by the same angle. It turns out that all isoclinic rotations can be described by either x —> p x (a left isoclinic rotation), or x —> x q (a right isoclinic rotation). The left isoclinic rotations form a group, isomorphic with S^3, as do also the right isoclinic rotations. And each of these groups is a *normal subgroup* of SO(4). Just like SO(n) with n even, n ≥ 6, the center of SO(4) is {-I, I}. But S(4)/{-I, I} is *not* a simple group. It is isomorphic to the cartesian product SO(3)×SO(3), and this is not simple because SO(3)×{I} and {I}×SO(3) are each a nontrivial closed normal subgroup. More at <https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space>. —Dan ————— * SO(n) is a group and a locally Euclidean topological space such that multiplication and inversion are continuous: i.e., it is a Lie group. Saying that a Lie group is "simple" means it has no nontrivial closed normal subgroup.