Around 1956, Boris Rosenfeld  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 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 . 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  and Freudenthal  found two very different versions of this construction in about 1958, but we shall start by presenting a simplified version published by E. Vinberg  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
This motivated Vinberg's definition of the magic square Lie
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
Yet another description of the magic square was recently given by Barton and Sudbery . 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
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  constructing and by means of spinors and rotation group Lie algebras:
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