half-lines from are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements. Beware: not all planes through are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-line are invariant. Assuming that a fixed orientation has been chosen for 4-dimensional space, isoclinic 4D rotations may be put into two categories. To see this, consider an isoclinic rotation , and take an orientation-consistent ordered set of mutually perpendicular half-lines at (denoted as ) such that and span an invariant plane, and therefore and also span an invariant plane. Now assume that only the rotation angle is specified. Then there are in general four isoclinic rotations in planes and with rotation angle , depending on the rotation senses in and . We make the convention that the rotation senses from to and from to are reckoned positive. Then we have the four rotations , , and . and are each other's inverses; so are and . As long as lies between 0 and , these four rotations will be distinct. Isoclinic rotations with like signs are denoted as left-isoclinic; those with opposite signs as right-isoclinic. Left- and right-isoclinic rotations are represented respectively by left- and right-multiplication by unit quaternions; see the paragraph "Relation to quaternions" below. The four rotations are pairwise different except if or . The angle corresponds to the identity rotation; corresponds to the central inversion, given by the negative of the identity matrix. These two elements of SO(4) are the only ones that are simultaneously left- and right-isoclinic. Left- and right-isocliny defined as above seem to depend on which specific isoclinic rotation was selected. However, when another isoclinic rotation with its own axes , , , is selected, then one can always choose the order of , , , such that can be transformed into by a rotation rather than by a rotation-reflection (that is, so that the ordered basis , , , is also consistent with the same fixed choice of orientation as , , , ). Therefore, once one has selected an orientation (that is, a system of axes that is universally denoted as right-handed), one can determine the left or right character of a specific isoclinic rotation. Group structure of SO(4) SO(4) is a noncommutative compact 6-dimensional Lie group. Each plane through the rotation centre is the axis-plane of a commutative subgroup