4.3 The Magic Square

Around 1956, Boris Rosenfeld [74] had the remarkable idea that just as is the isometry group of the projective plane over the octonions, the exceptional Lie groups , and are the isometry groups of projective planes over the following three algebras, respectively:

- the
**bioctonions**, , - the
**quateroctonions**, , - the
**octooctonions**, .

The situation is not so bad for the bioctonions:
is a simple Jordan algebra, though not a formally real one, and one can
use this to define
in a manner modeled after one
of the constructions of . Rosenfeld claimed that a similar
construction worked for the quateroctonions and octooctonions, but this
appears to be false. Among other problems,
and
do not become Jordan algebras under the product
. Scattered throughout the literature
[5,38,39] one can find frustrated comments
about the lack of a really nice construction of
and
. One problem is that these spaces do *not* satisfy the usual axioms for a projective plane. Tits addressed
this problem in his theory of 'buildings', which allows one to construct
a geometry having any desired algebraic group as symmetries
[90]. But alas, it still seems that the quickest way to get our
hands on the quateroctonionic and octooctonionic 'projective planes' is
by *starting* with the Lie groups and and then taking
quotients by suitable subgroups.

In short, more work must be done before we can claim to fully understand the geometrical meaning of the Lie groups and . Luckily, Rosenfeld's ideas can be used to motivate a nice construction of their Lie algebras. This goes by the name of the 'magic square'. Tits [89] and Freudenthal [37] found two very different versions of this construction in about 1958, but we shall start by presenting a simplified version published by E. Vinberg [92] in 1966.

First consider the projective plane when is a normed
division algebra . The points of this plane are the rank-1
projections in the Jordan algebra , and this plane admits a
Riemannian metric such that

Moreover, we have seen in equation (4.2) that

Combined with Rosenfeld's observations, these facts might lead one to hope that whenever we have a pair of normed division algebras and , there is a Riemannian manifold with

where for any -algebra we define

This motivated Vinberg's definition of the **magic square** Lie
algebras:

Now, when is commutative and associative, is a Lie algebra with the commutator as its Lie bracket, but in the really interesting cases it is not. Thus to make into a Lie algebra we must give it a rather subtle bracket. We have already seen the special case in equation (4.2). In general, the Lie bracket in is given as follows:

- and are commuting Lie subalgebras of .
- The bracket of with is given by applying to every entry of the matrix , using the natural action of as derivations of .
- Given
,

Here is the traceless part of the matrix , and given we define in the following way: is real-bilinear in and , and

where , , and are defined as in equation (4.1).

We will mainly be interested in the last row (or column), which is the
one involving the octonions. In this case we can take the magic square
construction as *defining* the Lie algebras , ,
and . This definition turns out to be consistent with our earlier
definition of .

Starting from Vinberg's definition of the magic square Lie algebras, we
can easily recover Tits' original definition. To do so, we need two
facts. First,

This is easily seen by direct examination of the relevant matrices. Second,

as vector spaces. This is just equation (4.2). Starting with Vinberg's definition and applying these two facts, we obtain

The last line is Tits' definition of the magic square Lie algebras. Unlike Vinberg's, it is not manifestly symmetrical in and . This unhappy feature is somewhat made up for by the fact that is a nice big Lie subalgebra. This subalgebra acts on in an obvious way, using the fact that any derivation of maps to itself, and any derivation of maps to itself. However, the bracket of two elements of is a bit of a mess.

Yet another description of the magic square was recently given by Barton and Sudbery [4]. This one emphasizes the role of trialities. Let be the Lie algebra of the group , where is the normed triality giving the normed division algebra . From equation (2) we have

To express the magic square in terms of these Lie algebras, we need
three facts. First, it is easy to see that

Second, Barton and Sudbery show that as vector spaces,

This follows in a case-by-case way from equation (3), but they give a unified proof that covers all cases. Third, they show that as vector spaces,

Now starting with Tits' definition of the magic square, applying the first two facts, regrouping terms, and applying the third fact, we obtain Barton and Sudbery's version of the magic square:

In the next three sections we use all these different versions of the magic square to give lots of octonionic descriptions of , and . To save space, we usually omit the formulas for the Lie bracket in these descriptions. However, the patient reader can reconstruct these with the help of Barton and Sudbery's paper, which is packed with useful formulas.

As we continue our tour through the exceptional Lie algebras, we shall make contact with Adams' work [1] constructing and by means of spinors and rotation group Lie algebras:

as vector spaces. Note that the numbers 9, 10, 12 and 16 are 8 more than the dimensions of and . As usual, this is no coincidence! In terms of the octonions, Bott periodicity implies that

This gives the following description of spinors in dimensions :

Table 6 — Spinor Representations Revisited

Since spinors in dimensions 1,2,4 and 8 are isomorphic to the division algebras and , spinors in dimensions 8 higher are isomorphic to the 'planes' and — and are thus closely linked to , , and , thanks to the magic square.

© 2001 John Baez