2 1. INTRODUCTION

Moufang quadrangles from which they arise, thereby providing a new proof for the

classification of Moufang quadrangles, which does consist of the division into parts

(A) and (B).

The Moufang hexagons all arise from forms of algebraic groups of type G2,

3Z)4,

EQ or E$, or they are of mixed type associated with groups of type G2. The

Moufang quadrangles arise either from certain classical groups or from forms of

algebraic groups of type EQ , Ej or E% or are of mixed type associated with groups of

type B2 or F4. The quadrangular systems parametrize the Moufang quadrangles in

the same way that the Jordan algebras mentioned above parametrize the Moufang

hexagons, and it is our hope that the quadrangular systems will turn out to be

equally interesting objects of study.

We start by giving the (ad hoc) definition of the quadrangular systems. Consid-

ering the background of the Moufang quadrangles, it should not be too surprising

that we need a large number of axioms to describe these systems. In the next

chapter, we give some elementary properties of these systems. In chapter 4, we

explain how to construct a Moufang quadrangle starting from an arbitrary quad-

rangular system. In chapter 5, we show that every Moufang quadrangle arises in

this way. After a couple of remarks in chapter 6, we present a list of 6 examples

of quadrangular systems, which corresponds to the 6 different classes of Moufang

quadrangles as described in [20]. Finally, chapter 8 is devoted to the classification

of the quadrangular systems. We conclude with an appendix in which we restate

the axiom system for abelian quadrangular systems and for some specific subclasses

of those.

Acknowledgment

I am very grateful to Richard Weiss, for providing me a copy of his book "Mou-

fang Polygons" [20] prior to publication, and for the many illuminating discussions

we have had on this topic.