Fourfolds of Weil type and the spinor map

Recent papers by Markman and O'Grady give, besides their main results on the Hodge conjecture and on hyperkahler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.