In § 11.3, we discussed the transformations that preserve the determinant and trace of Jordan matrices $\XXX\in\bH_3(\OO)$; such transformations are elements the group $F_4=\SU(3,\OO)$. It is now straightforward to remove the restriction on the trace, and consider all transformations on $\bH_3(\OO)$ that preserve the determinant; this group would deserve the name $\SL(3,\OO)$.

Again, we make use of our knowledge of the $2\times2$ case, in this case the transition from $\SU(2,\OO)\cong\Spin(9)$ to $\SL(2,\OO)\cong\Spin(9,1)$, as discussed in § 9.2 and § 9.3, This transition involves the addition of nine boosts, and there are three natural embeddings of $2\times2$ matrices inside $3\times3$ matrices (“types”), so we expect to add $3\times9=27$ boosts to $\SU(3,\OO)$ in order to get $\SL(3,\OO)$. However, three of these boosts are diagonal matrices, namely the boosts with “$z$”, and only two of these are independent. Thus, $\SL(3,\OO)$ contains $52+26=78$ elements, that is, \begin{equation} |\SL(3,\OO)| = 78 \end{equation} This group is another of the exceptional Lie groups, and is classified as $E_6$. More precisely, $\SL(3,\OO)$ is a particular real form of $E_6$, namely the one with $52$ rotations and $26$ boosts; its signature is therefore $26-52=-26$, and we write \begin{equation} E_{6(-26)} = \SL(3,\OO) \end{equation}

For completeness, we list the 78 elements of $\SL(3,\OO)$ here explicitly. Since \begin{equation} \SU(3) \subset G_2 \subset \SO(7) \subset \SO(8) \subset \SU(2,\OO) \subset F_4 \subset E_{6(-26)} \end{equation} all of these groups are subgroups of $\SL(3,\OO)$, as is $\SL(2,\OO)$. the elements of these subgroups are also identified below.

• As discussed in § 11.1, the $\SU(3)\subset G_2$ that fixes $\ell$ is generated by the eight elements $\{A_i$, $A_j$, $A_k$, $A_{k\ell}$, $A_{j\ell}$, $A_{i\ell}$, $A_\ell$, $G_\ell\}$.
• The six remaining generators of $G_2$ are $\{G_i$, $G_j$, $G_k$, $G_{k\ell}$, $G_{j\ell}$, $G_{i\ell}\}$. 1)
• The seven remaining generators of (type I) $\SO(7)$ are $\{S_i$, $S_j$, $S_k$, $S_{k\ell}$, $S_{j\ell}$, $S_{i\ell}$, $S_\ell\}$.
• The seven remaining generators of $\SO(8)$ are $\{D_i$, $D_j$, $D_k$, $D_{k\ell}$, $D_{j\ell}$, $D_{i\ell}$, $D_\ell\}$.
• The eight remaining generators of $\SU(2,\OO)\cong\SO(9)$ are $\{R^I_1$, $R^I_i$, $R^I_j$, $R^I_k$, $R^I_{k\ell}$, $R^I_{j\ell}$, $R^I_{i\ell}$, $R^I_\ell\}$.
• The 16 remaining generators of $F_4$ are $\{R^{II}_1$, $R^{II}_i$, $R^{II}_j$, $R^{II}_k$, $R^{II}_{k\ell}$, $R^{II}_{j\ell}$, $R^{II}_{i\ell}$, $R^{II}_\ell\}$ and $\{R^{III}_1$, $R^{III}_i$, $R^{III}_j$, $R^{III}_k$, $R^{III}_{k\ell}$, $R^{III}_{j\ell}$, $R^{III}_{i\ell}$, $R^{III}_\ell\}$.
• The 9 boosts needed to get from $\SU(2,\OO)$ to $\SL(2,\OO)\cong\SO(9,1)$ were given (as $2\times2$ matrices) in § 9.3. Rewritten as $3\times3$ matrices, they take the form \begin{equation} B^I_q = \begin{pmatrix} \cosh\alpha& \bar{q}\sinh\alpha& 0\\ q\sinh\alpha& \cosh\alpha& 0\\ 0& 0& 1\\ \end{pmatrix} \label{boostsI} \end{equation} for $q=\{1,i,j,k,k\ell,j\ell,i\ell,\ell\}$, as well as \begin{equation} B^I_z = \begin{pmatrix} e^\alpha& 0& 0\\ 0& e^{-\alpha}& 0\\ 0& 0& 1\\ \end{pmatrix} \label{zboost} \end{equation}
• Finally, the remaining 17 generators of $E_6$ are the 16 boosts $B^{II}_q$ and $B^{III}_q$ obtained by cyclically permuting (\ref{boostsI}), together with the single independent boost obtained by cyclically permuting (\ref{zboost}), which we take to be \begin{equation} B_3 = \begin{pmatrix} e^\alpha& 0& 0\\ 0& e^{\alpha}& 0\\ 0& 0& e^{-2\alpha}\\ \end{pmatrix} \end{equation}

We have exhibited an explicit set of $8+6+7+7+8+16+9+17=78$ generators for $E_6$, as claimed.