An earlier theorem, proved independently by Tits and Weiss,[2][3] showed that a Moufang polygon must be a generalized 3-gon, 4-gon, 6-gon, or 8-gon, so the purpose of the aforementioned book was to analyze these four cases.
Real forms of Lie groups give rise to examples which are the three main types of Moufang 3-gons.
The fourth case — a form of E6 — is exceptional, and its analogue for Moufang 4-gons is a major feature of Weiss's book.
The third case involves "alternative" division algebras (which satisfy a weakened form of the associative law), and a theorem of Richard Bruck and Erwin Kleinfeld[4] shows that these are Cayley-Dickson algebras.
They can be divided into three classes: There is some overlap here, in the sense that some classical groups arising from pseudo-quadratic spaces can be obtained from quadrangular algebras (which Weiss calls special), but there are other, non-special ones.
These are the most exotic of all—they involve purely inseparable field extensions in characteristic 2—and Weiss only discovered them during the joint work with Tits on the classification of Moufang 4-gons by investigating a strange lacuna that should not have existed but did.
The classification of Moufang 4-gons by Tits and Weiss is related to their intriguing monograph in two ways.
In fact all the exceptional Moufang planes, quadrangles, and hexagons that do not arise from "mixed groups" (of characteristic 2 for quadrangles or characteristic 3 for hexagons) come from octonions, quadrangular algebras, or Jordan algebras.