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].