In this talk, I will argue why a Clifford algebraic framework is ideally suited for describing reflection groups and root systems. This approach resulted in an induction theorem relating sets of root systems in three and four dimensions, with connections to Arnold's Trinities and the McKay correspondence. This connection can be extended - by matching invariants - to different ADE-type correspondences, with a conceptual unification between the different ADE sets at the level of root systems.