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This paper uses rigid Hecke eigensheaves, building on Yun's work on theconstruction of motives with exceptional Galois groups, to produce the firstrobust examples of `generalized Kuga-Satake theory' outside the Tannakiancategory of motives generated by abelian varieties. To strengthen ourdescription of the `motivic' nature of Kuga-Satake lifts, we digress toestablish a result that should be of independent interest: for anyquasi-projective variety over a (finitely-generated) characteristic zero field,the associated weight-graded of its intersection cohomology arises from amotivated motive in the sense of Andr\'{e}, and in particular from a classicalhomological motive if one assumes the Standard Conjectures. This extends workof de Cataldo and Migliorini.

Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves