We showed in § 9.1.1 that conjugation by a unit-normed imaginary quaternion $u$ yields a flip of imaginary quaternions about the $u$-axis. Flips can not only be used with quaternions, but also with octonions, since the expression $px\bar{p}$ involves only two directions, and hence lies in a quaternionic subalgebra of $\OO$; there are no associativity issues here.

But any rotation can be constructed from two flips. For instance, to rotate the $xy$-plane, first pick any line (through the origin) in that plane. Now rotate by $\pi$ about that line, that is, take all points in the plane orthogonal to the given line by $-1$. For example, if the chosen line is the $x$-axis, then the $x$-coordinate of a point is unaffected, while its $y$- and $z$-coordinates are multiplied by $-1$. Now pick another line in the $xy$-plane, at an angle $\alpha$ from the first line, and repeat the process. Points along the $z$ axis are reflected twice, and are thus taken back to where they started. But any point in the $xy$-plane winds up being rotated by $2\alpha$! (This is easiest to see for points along the $x$-axis.)

It doesn't matter which two lines in the $xy$-plane we choose, so long as they are separated by $\alpha$ (with the correct orientation). And we have described this procedure as though it were taking place in three dimensions, but in fact it works in any number of dimensions; there can be any number of “$z$-coordinates”, all of which are flipped twice, and return to where they started.

To rotate counterclockwise by an angle $2\alpha$ in the $ij$-plane, we therefore begin by conjugating with $i$, thus rotating about the $i$-axis. To complete the $ij$ rotation, we need to rotate about the line in the $ij$-plane which makes an angle $\alpha$ with the $i$-axis. This is accomplished by conjugating by a unit octonion $u$ pointing along the line, which is easily seen to be \begin{equation} u = i \cos\alpha + j \sin\alpha \label{sflip} \end{equation} Finally, note that the conjugate of any imaginary octonion is just minus itself. Putting this all together, a rotation by $2\alpha$ in the $ij$-plane is given by \begin{equation} x \longmapsto (i\cos\alpha+j\sin\alpha)(ixi)(i\cos\alpha+j\sin\alpha) \label{ijflip} \end{equation} for any octonion $x$. (We have removed two minus signs.)

If $x\in\HH$, we can collapse the parentheses in (\ref{ijflip}), obtaining \begin{equation} x \longmapsto (-\cos\alpha-k\sin\alpha)x(-\cos\alpha+k\sin\alpha) = e^{k\alpha} x e^{-k\alpha} \end{equation} which is just conjugation by a unit quaternion, as in the construction of $\SO(3)$ in §9.1.1. Over the octonions, however, we can not simplify (\ref{ijflip}) any further; it takes two transformations, not just one, to rotate a single plane. We refer to this process as nesting, and describe the transformaion (\ref{ijflip}) as a nested flip.

We can repeat this construction using any unit-normed imaginary units $u$, $v$ that are orthogonal to each other, obtaining the rotation in the $uv$-plane. Since such rotations generate $\SO(7)$, we have \begin{align} \SO(7) &= \langle \{ x \longmapsto (u\cos\alpha+v\sin\alpha)(uxu)(u\cos\alpha+v\sin\alpha) : \nonumber\\ &\qquad u,v,x\in\OO, u^2=-1=v^2, \{u,v\}=0 \} \rangle \end{align}