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, 3 Z)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.

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