Octonionic SO(7)

Section 2.6 Octonionic \(SO(7)\)

(The material in this section was originally published in [1].)

We showed in Section 2.3 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 \(\OO\text{;}\) 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, which is the same as reflecting points through the line. 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 \(-1\text{.}\) 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 \(2\alpha\text{!}\) (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 \(i\text{,}\) 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\tag{2.6.1} \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)\tag{2.6.2} \end{equation}

for any octonion \(x\text{.}\) (We have removed two minus signs.)

If \(x\in\HH\text{,}\) we can collapse the parentheses in (2.6.2), obtaining

\begin{equation} x \longmapsto (-\cos\alpha-k\sin\alpha)x(-\cos\alpha+k\sin\alpha) = e^{k\alpha} x e^{-k\alpha}\tag{2.6.3} \end{equation}

which is just conjugation by a unit quaternion, as in the construction of \(\SO(3)\) in Section 2.3. Over the octonions, however, we can not simplify (2.6.2) 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 (2.6.2) as a nested flip.

We can repeat this construction using any unit-normed imaginary units \(u\text{,}\) \(v\) that are orthogonal to each other, obtaining the rotation in the \(uv\)-plane. Since such rotations generate \(\SO(7)\text{,}\) we have

\begin{align} \SO(7) \amp= \langle \{ x \longmapsto (u\cos\alpha+v\sin\alpha)(uxu)(u\cos\alpha+v\sin\alpha) :\notag\\ \amp\qquad u,v,x\in\OO, u^2=-1=v^2, \{u,v\}=0 \} \rangle\tag{2.6.4} \end{align}