I have a question about a particular type of decomposition of N-dimensional rotation matrices. The specific types of matrices I am considering are orthogonal matrices (the rows form a set of orthogonal unit vectors, as do the columns). Further, I am only considering the case where the determinant is 1 (as opposed to -1). I.e., I am disallowing reflections. I am interested in factoring such a matrix into a sequence of "primitive rotations" (I'm not sure what the proper term is). By this I mean rotations which only affect two coordinate axes. This defines a rotation within a plane, with the restriction that the plane be the product of two of the standard coordinate axes. For an N-dimensional matrix, there are (N take 2) = N*(N-1)/2 possible pairs of axes. For example, in 3 dimensions they are XY, XZ, and YZ. Any such rotation matrix, as defined above, can be factored into a sequence of N*(N-1)/2 primitive rotations, which each pair of axes occurring exactly once. I know certain orderings will work. One such ordering is to first perform all rotations involving some particular axis, then perform all remaining rotations involving some other particular axis, etc. For instance, in 4 dimensions, the sequence could be (a1,a2), (a1,a3), (a1,a4), (a2,a3), (a2,a4), (a3,a4). The opposite ordering can easily be shown to work as well, i.e. (a3,a4), (a2,a4), (a2,a3), (a1,a4), (a1,a3), (a1,a2). My question is this: Can *any* fixed ordering be made to work, for any such rotation matrix? Or do only some orderings work? Some observations: There are a number of transformations that can be performed on any ordering to produce a new ordering which will work. In some cases the rotation amounts can be preserved. In others, the rotation amounts need to change. Here's an example of a transformation that works in 4 dimensions: (a1,a2), (a1,a3), (a2,a3), (a1,a4), (a2,a4), (a3,a4) Here I have have swapped adjacent primitive rotations (a1,a4) and (a2,a3). This can always be done, without changing the rotation amounts, since they have no coordinates in common. But what about the case where they do have a coordinate in common? In three dimensions, we know that any order works, so XY, XZ, YZ works, and so does XY, YZ, XZ, although when switching from one to the other, in general all *three* rotation amounts must be changed. Does anyone know the answer to this? In N dimensions, will any fixed ordering work? Or do only some fixed orderings work? Tom