The Freudenthal–Tits magic square was originally given in terms of Lie algebras; the version shown in Table 3 lists instead particular real forms of the corresponding Lie groups. Vinberg later gave a symmetric parameterization of the Lie algebras $\alg_3(\KK_1,\KK_2)$ in this magic square in the form \begin{equation} \alg_3(\KK_1,\KK_2) = sa(3,\KK_1\otimes\KK_2) \oplus \der(\KK_1) \oplus \der(\KK_2) \label{Vin3} \end{equation} where $\KK_1$, $\KK_2$ are division algebras (or possibly their split cousins), $sa(n,\KK_1)$ denotes the anti-Hermitian tracefree $n\times n$ matrices with elements in $\KK_1$ and $\der(\KK_1)$ denotes the derivations of $\KK_1$, which are the Lie algebra version of automorphisms. The division algebras $\RR$ and $\CC$ have no derivations, and \begin{align} \der(\HH) &= \so_3 \\ \der(\OO) &= \gg_2 \end{align} Thus, the only change in the Vinberg construction between the $2\times2$ and $3\times3$ cases is the use of $\gg_2$ rather than $\so_7$ in the octonionic case, a difference that can be attributed to the lack of triality in the $2\times2$ case.

This magic square is indeed symmetric; corresponding groups are again different real forms of the same group. But the most remarkable property of the Freudenthal–Tits magic square is that it contains four of the five exceptional Lie groups—and the fifth, $G_2$, appears implicitly as part of the Vinberg construction.

The Vinberg construction also suggests that the groups in the Freudenthal–Tits magic square can be described as $\SU(3,\KK'\otimes\KK)$, by analogy with the $2\times2$ case considered in § 15.2. This construction is indeed possible, but is beyond the scope of this book.

Table 3: The “half-split” $3\times3$ magic square of Lie groups.