Orthogonal Projections of Root Systems to a Plane

Consider the type Formula: see text root system with simple roots Formula: see text, and let Formula: see text be the span of Formula: see text and Formula: see text over Formula: see text. Let Formula: see text be any irreducible reduced root system but not of type Formula: see text, with rank Formula: see text. Namely, Formula: see text is of type Formula: see text (Formula: see text), Formula: see text (Formula: see text), Formula: see text (Formula: see text), Formula: see text, Formula: see text, Formula: see text or Formula: see text. We show that the set of nonzero orthogonal projection vectors of Formula: see text to the plane Formula: see text forms the inverse root system of Formula: see text.