Automorphism groups of root system matroids

Given a root system R, the vector system R̃ is obtained by taking a representative v in each antipodal pair {v,−v}. The matroid M(R) is formed by all independent subsets of R̃. The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root systems matroids M(R) are uniquely determined by their independent sets of size 3. As a corollary, we compute these groups explicitly, and thus complete the classification of the automorphism groups of root systems matroids.