As suggested in § 11.2, the Albert algebra $\bH_3(\OO)$ can be regarded as a generalization of the vector space $\RR^3$ to the octonions. The elements $\XXX\in\bH_3(\OO)$ of the Albert algebra are Hermitian matrices, and have well-defined determinants. We therefore seek transformations that preserve these properties.

Reasoning by analogy with § 7.2, we seek transformations of the form \begin{equation} \XXX \longmapsto \MMM\XXX\MMM^\dagger \label{MXM3} \end{equation} that preserve the determinant and trace of $\XXX$; this group would deserve the name $\SU(3,\OO)$.

Before proceeding further, however, we must ask under what circumstances $\MMM\XXX\MMM^\dagger$ is well-defined, that is, when is it true that \begin{equation} (\MMM\XXX)\MMM^\dagger = \MMM(\XXX\MMM^\dagger) \end{equation} As in § 9.3, we therefore assume that the elements of $\MMM$ lie in some complex subalgebra of $\OO$, which can be different for different $\MMM$, and we will need to allow nesting, that is, not all transformations can be carried out with a single $\MMM$.

In § 9.2, we showed that $\SU(2,\OO)\cong\SO(9)$. There are three obvious ways to embed $2\times2$ matrices inside $3\times3$ matrices, depending on which row and column are ignored; we refer to these alternatives as types. Thus, $\SU(3,\OO)$ is contained in the union of the three copies of $\SU(2,\OO)$, each of which has $36$ elements. However, there is substantial overlap between these three subgroups.

Recall from § 9.1.4 that left, right, and symmetric multiplication over $\OO$ all yield representations of $\SO(8)$, related by triality. Consider now the three off-diagonal elements of a Jordan matrix $\XXX$, each of which is of course an element of $\OO$. Consider further the diagonal elements of $\SU(2,\OO)$, embedded (in three ways) in $\SU(3,\OO)$, acting on $\XXX$; such elements take the form 1) \begin{equation} \MMM = \begin{pmatrix} p& 0& 0\\ 0& p& 0\\ 0& 0& 1\\ \end{pmatrix} \label{pconj} \end{equation} with $p^2=-1$, or \begin{equation} \MMM = \begin{pmatrix} q& 0& 0\\ 0& \bar{q}& 0\\ 0& 0& 1\\ \end{pmatrix} \label{qsym} \end{equation} where $|q|=1$, or cyclic permutations of these matrices. Under (\ref{MXM3}), each such transformation leaves the diagonal of $\XXX$ alone, and acts separately on the three off-diagonal elements, in each case implementing one of the three representations of $\SO(8)$.

We conclude that the three copies of $\SO(8)$, coming from the three copies of $\SU(2,\OO)$ embedded in $\SU(3,\OO)$, must in fact be the same, which is an important consequence of triality. Redoing our count, we now have a single copy of $\SO(8)$, with its 28 elements. However, the eight remaining transformations coming from each copy of $\SU(2,\OO)$ are indeed different (since they act on different pairs of diagonal elements), so we have $28+3\times8=52$ elements in all, that is, \begin{equation} |\SU(3,\OO)| = 52 \end{equation} This group is one of the exceptional Lie groups, and is classified as $F_4$, that is, \begin{equation} F_4 = \SU(3,\OO) \end{equation}

We digress briefly to discuss some further properties of triality, which plays an essential role in the above description of $F_4$. Including cyclic permutations, there are only 14 independent transformations of the form (\ref{qsym}), since the third set of seven can be constructed from the other two. These 14 transformations act on the three octonions in $\XXX$, acting on one octonion by symmetric multiplication, another by left multiplication, and the third by right multiplication. Each of these transformations is in $\SO(8)$, but they are different from each other, related by triality. As for the remaining transformations, of the form (\ref{pconj}), there appear to be only seven of them, but there are really ${7\choose2}=21$, due to nesting. However, seven of these can also be represented using (cyclically permuted) transformations of the form (\ref{qsym}); the remaining 14 transformations, which can only be represented using nesting, are precisely the generators of $G_2$. In this case, and only in this case, symmetric, left, and right multiplication yield the same transformation in $\SO(8)$, a property that we refer to as strong triality. An example of strong triality is the identity \begin{equation} -i(j(kxk)j)i = -i(j(kx)) = ((xk)j)i \end{equation} for any $x\in\OO$ (where the minus signs are needed due to the odd number of factors in this nonstandard instance of nesting).

We conclude this section by listing an explicit set of 52 generators for $F_4$.

• There are seven independent elements of the form (\ref{pconj}), which when nested yield the ${7\choose2}=14$ generators of $G_2$. Since a rotation in a single plane is represented by two nested flips, a typical $G_2$ transformation involving two such rotations requires four nested flips. An example of such a $G_2$ transformation is \begin{equation} \XXX\longmapsto\MMM_4\left(\MMM_3\left(\MMM_2\left(\MMM_1\XXX \MMM_1^\dagger\right)\MMM_2^\dagger\right)\MMM_3^\dagger\right)\MMM_4^\dagger \end{equation} where \begin{align} \MMM_1 &= \begin{pmatrix} i& 0& 0\\ 0& i& 0\\ 0& 0& 1\\ \end{pmatrix} \\ \MMM_2 &= \begin{pmatrix} p& 0& 0\\ 0& p& 0\\ 0& 0& 1\\ \end{pmatrix} \\ \MMM_3 &= \begin{pmatrix} i\ell& 0& 0\\ 0& i\ell& 0\\ 0& 0& 1\\ \end{pmatrix} \\ \MMM_4 &= \begin{pmatrix} q& 0& 0\\ 0& q& 0\\ 0& 0& 1\\ \end{pmatrix} \end{align} with $p=i\cos\alpha+j\sin\alpha$ and $q=i\ell\cos\alpha+j\ell\sin\alpha$. In § [[book:go:g2|11.1], we referred to this transformation (acting on $\OO$) as $A_k$.
• There are seven independent elements of the form (\ref{qsym}). An example of such an $\SO(8)$ transformation is \begin{equation} \MMM = \begin{pmatrix} e^{i\alpha}& 0& 0\\ 0& e^{-i\alpha}& 0\\ 0& 0& 1\\ \end{pmatrix} \end{equation} acting via (\ref{MXM3}). We will refer to this transformation as $D_i$.
• The remaining seven independent elements of $\SO(8)$ can be chosen to take the form \begin{equation} \MMM = \begin{pmatrix} e^{i\alpha}& 0& 0\\ 0& e^{i\alpha}& 0\\ 0& 0& e^{-2\alpha}\\ \end{pmatrix} \end{equation} again acting via (\ref{MXM3}). We will refer to this transformation as $S_i$.
• There are an additional eight elements of “type I” $\SO(9)$, which take the form \begin{equation} \MMM = \begin{pmatrix} \cos\alpha& -\bar{q}\sin\alpha& 0\\ q\sin\alpha& \cos\alpha& 0\\ 0& 0& 1\\ \end{pmatrix} \end{equation} where $q=\{1,i,j,k,k\ell,j\ell,i\ell,\ell\}$, together with $2\times8=16$ further elements obtained from these by cyclic permutations (“types II and III”). We will refer to these transformations as $R^I_q$, $R^{II}_q$, and $R^{III}_q$, respectively.

We have exhibited an explicit set of $14+7+7+3\times8=52$ generators for $F_4$, as claimed.