THE GEOMETRY OF THE EXCEPTIONAL LIE GROUPS

Section 7.3 The \(3\times3\) Magic Square

The half-split \(3\times3\) magic square of Lie algebras, also known as the (half-split) Tits–Freudenthal magic square, is shown in Table 7.3.1. Some of the algebras are labeled by their common names; the remaining algebras are labeled by their Cartan–Killing type and their rank \(m\) (the dimension of the Cartan subalgebra), and by their signature \(s=b-r\text{,}\) the difference between the number of boosts and rotations. For example, the split real form \(\ff_{4(4)}\) has a Cartan subalgebra of dimension \(4\text{,}\) consisting entirely of boosts; the remaining elements divide evenly into rotations and boosts. Once one knows that \(|\ff_4|=52\text{,}\) it is then easy to deduce that \(\ff_{4(4)}\) contains 28 boosts and 24 rotations.

But where did these algebras come from?

As discussed in Section 7.2, we can extend the construction of the \(2\times2\) magic square of orthogonal Lie algebras to the \(3\times3\) case, thus extending the description of the former as \(\su(2,\KK'\otimes\KK)\) to a construction of \(\su(3,\KK'\otimes\KK)\text{.}\) As in the \(2\times2\) case, the subtlety is in the handling of the diagonal matrices.

We begin with the observation that \(\su(2,\KK'\otimes\KK)\) can be constructed by starting with the off-diagonal elements, as these elements generate all of the remaining elements. Explicitly, the element

corresponds to a rotation in the \(aA\)-plane, assuming that \(a\in\OO\) and \(A\in\OO'\) are unit octonions. Composition of such rotations yields the remaining rotations within both \(\OO\) and \(\OO'\text{.}\) In the \(2\times2\) case, this process led us to include imaginary multiples of the identity when working over \(\HH\) and/or \(\HH'\text{,}\) and to include nested products of imaginary multiples of the identity when working over \(\OO\) and/or \(\OO'\text{.}\)

In the \(3\times3\) case, the same general principles hold. We can start with the three independent off-diagonal elements, each of which contains an element of \(\KK'\otimes\KK\text{,}\) and compose the action of all such matrices. Again, this yields some diagonal matrices; the question is, how many.

Over the quaternions, the answer is a straightforward generalization of the \(2\times2\) case, requiring three additional imaginary diagonal matrices to incorporate the extra diagonal degree of freedom. However, over the octonions, there is a surprise: Some of the nested actions we had in the \(2\times2\) case can be generated instead by unnested \(3\times3\) actions.

To make a long story short, the automorphism group of the quaternions is the same as the rotation group on the imaginary quaternions; both are \(\SO(3)\text{,}\) with three generators. Over the octonions, however, the automorphism group is \(\gg_2\text{,}\) with 14 generators, which is a subgroup of the rotation group \(\SO(7)\) on the imaginary octonions, which has 21 generators. So rather than having to add 21 nested flips, we only need 14.

Bearing these considerations in mind, it is reasonably straightforward to construct the Lie algebras \(\su(3,\KK'\otimes\KK)\) in all 16 cases–and in all three magic squares, namely split, half-split, and compact. The resulting algebras can be classified, yielding, in the half-split case, Table 7.3.1.

It is remarkable that such a parameterization of Lie algebras in terms of (possibly split) division algebras exists, and even more that it includes all of the exceptional cases—except \(\gg_2\text{,}\) which nonetheless plays a key role. These exceptional Lie algebras are discussed in more detail in both [1] and Chapter 8.

No further generalization is possible. Although matrix algebras of any dimension form a Lie algebra over \(\RR\text{,}\) \(\CC\text{,}\) or \(\HH\text{,}\) the lack of associativity prevents the construction of sensible matrix algebras over \(\OO\) of dimension greater than \(3\times3\text{.}\)