Abstract: Markman and O'Grady uncovered a deep relation between abelian fourfolds of Weil type with discriminant 1 and hyper-Kähler varieties of generalized Kummer type, at the level of Hodge theory and period domains. Markman was able to use this to prove the Hodge conjecture for these fourfolds; he later found also a different proof which works for Weil fourfolds with arbitrary discriminant. In my talk I will explain how Weil fourfolds with discriminant 1 are very closely related to certain hyper-Kähler varieties of OG6-type, in a direct and geometric way. As a consequence, we obtain another proof of the Hodge conjecture for Weil fourfolds with discriminant 1, as well as for many families of hyper-Kähler varieties of OG6-type which form loci of codimension 1 in their moduli spaces. The results that I will discuss are joint work with Lie Fu.