In 1977 Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian varieties are said to be of Weil type and these Hodge classes are known as Weil classes. We prove that the Weil classes are algebraic for all abelian sixfold of Weil type of discriminant -1, for all imaginary quadratic number fields. The algebraicity of the Weil classes follows for all abelian fourfolds of Weil type (for all discriminants and all imaginary quadratic number fields), by a degeneration argument of C. Schoen. The main two ingredients of our proof are:

1) A natural equivalence, due to Orlov, between the cartesian square XxX of an abelian n-fold X and XxPic^0(X), inducing an isomorphism of cohomologies equivariant with respect to the natural action of the derived monodromy group Spin(2n,2n) of X. Orlov's equivalence maps the tensor product of two suitably chosen (secant) sheaves on X to an object E on XxPic^0(X) with a characteristic class invariant under the Mumford-Tate group of abelian varieties of Weil type.

2) When X is the Jacobian of a genus 3 curve, we prove that the object E on XxPic^0(X) is semi-regular and use the Buchweitz-Flenner semi-regularity theorem in order to deform it to deformation equivalent abelian sixfolds of Weil type.