  Given a root system $\mathsf{R}$, the vector system $\tilde{\mathsf{R}}$ is
obtained by taking a representative $v$ in each antipodal pair $\{v, -v\}$. The
matroid $M(\mathsf{R})$ is formed by all independent subsets of
$\tilde{\mathsf{R}}$. 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 $M(\mathsf{R})$ 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.
