In § 9.2, we showed that the groups $\SU(2,\KK)$ are \begin{align} \SU(2,\RR)&\cong\SO(2) \\ \SU(2,\CC)&\cong\SO(3) \\ \SU(2,\HH)&\cong\SO(5) \label{su2h}\\ \SU(2,\OO)&\cong\SO(9) \end{align} for $\KK=\RR,\CC,\HH,\OO$, respectively (and where all congruences are local, that is, up to double cover). Similarly, in § 9.3, we showed that the groups $\SL(2,\KK)$ are \begin{align} \SL(2,\RR)&\cong\SO(2,1) \\ \SL(2,\CC)&\cong\SO(3,1) \\ \SL(2,\HH)&\cong\SO(5,1) \\ \SL(2,\OO)&\cong\SO(9,1) \end{align} Finally, combining results from § 7.5, § 8.2, and § 9.4, we showed that \begin{align} \Sp(4,\RR) &\cong \SU(2,\HH') \cong \SO(3,2) \label{su2hp}\\ \Sp(4,\CC) &\cong \SU(2,2) \cong \SO(4,2) \end{align} There is a pattern here, which correctly suggests that \begin{align} \Sp(4,\HH) &\cong \SO(6,2) \\ \Sp(4,\OO) &\cong \SO(10,2) \end{align} as shown in Table 1. This table has some remarkable properties. Looking at the three columns labeled by $\RR$, $\CC$, and $\HH$, there is a symmetry between rows and columns. Corresponding groups, such as $\SO(3)$ and $\SO(2,1)$, have the same dimension, and are in fact merely different real forms of the same group. Furthermore, comparing (\ref{su2h}) with (\ref{su2hp}) leads us to suspect, again correctly, that the rows can perhaps be labeled with the split division algebras.

These symmetries lead us to suspect that there should be a fourth row in Table 1, and a unified description of all of these groups at once. We provide such a description in § 15.2.

Table 1: The orthogonal, Lorentz, and symplectic groups over the division algebras.