In the classification of Moufang polygons by J. Tits and R. Weiss, the most intricate case is by far the case of the exceptional Moufang quadrangles of type
E
6
E_6
,
E
7
E_7
and
E
8
E_8
, and in fact, the construction that they present is ad-hoc and lacking a deeper explanation. We will show how tensor products of two composition algebras can be used to construct these Moufang quadrangles in characteristic different from 2.
As a byproduct, we will obtain a method to construct any Moufang quadrangle in characteristic different from 2 from a module for a Jordan algebra.