In higher dimensions, not every rotation corresponds to a single axis together with an angle of rotation about this axis; the concept of Euler angles does not generalize. The first problem is that, even for rotations in a given plane, it is not possible to associate an “axis” with a given rotation, as there are multiple directions orthogonal to the plane of rotation. Furthermore, it is no longer the case that every rotation corresponds to rotation in a single plane.

We first encounter this difficulty in four dimensions, where we can rotate each of two independent planes arbitrarily. For this reason, we make no effort to identify all matrices satisfying the definition \begin{equation} \SO(4) = \{ M\in\RR^{4\times 4} : M^T M = I, \det M = 1 \} \end{equation} but rather rely on the fact that any such rotation can be generated by rotations in coordinate planes. How many such planes are there? In $n$ dimensions, there are ${n\choose2}$ possible planes, so the dimension of the orthogonal groups is given by \begin{equation} |\SO(n)| = \frac12 n(n-1) \end{equation} In four dimensions, there are ${4\choose2}=6$ possible planes; there are six matrices of the same general form as our $R_x$, $R_y$, $R_z$ from $\SO(3)$. In $\SO(3)$, however, we labeled our generators by the axis about which they rotate; in higher dimensions, we must instead label them by the plane in which they rotate. So start by relabeling our generators of $\SO(3)$ as $R_{yz}$, $R_{zx}$, $R_{yz}$, and reinterpret them as transformations in four dimensions that hold the fourth axis, $w$ say, fixed. Then the three remaining generators of $\SO(4)$ are $R_{wx}$, $R_{wy}$, $R_{wz}$, and we have \begin{equation} \SO(4) = \langle R_{yz},R_{zx},R_{yz},R_{wx},R_{wy},R_{wz} \rangle \end{equation}

Each of these generators of $\SO(4)$ corresponds to a rotation in a single plane, leaving the orthogonal plane invariant. We can instead consider rotations that rotate two orthogonal planes. Although the angles of rotation in the two planes could be different, we consider the special case where these angles are equal in magnitude. Such rotations are called isoclinic. Consider therefore the transformations \begin{align} S^\pm_{yz} &= S^\pm_{yz}(\alpha) = R_{yz}(\alpha)R_{wx}(\pm\alpha) \\ S^\pm_{zx} &= S^\pm_{zx}(\alpha) = R_{zx}(\alpha)R_{wy}(\pm\alpha) \\ S^\pm_{xy} &= S^\pm_{xy}(\alpha) = R_{xy}(\alpha)R_{wz}(\pm\alpha) \end{align} and the subsets \begin{equation} SO4^\pm = \langle S^\pm_{yz},S^\pm_{zx},S^\pm_{xy} \rangle \end{equation} of $\SO(4)$. Remarkably, each of these subsets closes under multiplication; $SO4^\pm$ are subgroups of $\SO(4)$. It is not hard to see that the multiplication table for the generators of $SO4^\pm$ is identical to that of $\SO(3)$; we say that \begin{equation} SO4^\pm \cong \SO(3) \end{equation} (“each of $SO4^\pm$ is isomorphic to $\SO(3)$”). Furthermore, we can recover all of $\SO(4)$ by multiplying elements of these two subgroups together. For instance, \begin{align} S^+_{xy}(\alpha) S^-_{xy}(\alpha) &= R_{xy}(2\alpha) \\ S^+_{xy}(\alpha) S^-_{xy}(-\alpha) &= R_{wx}(2\alpha) \end{align} and we have shown that \begin{equation} \SO(3)\times\SO(3) = \SO(4) \end{equation} a result that we will revisit in quaternionic language in §9.1.2.