Let us recall that a catalogue of simplicial hyperplane arrangements was firstly provided by Grünbaum in 1971 [B. Grünbaum, in: Proc. 2nd Louisiana Conf. Combin., Graph Theory, Computing; Baton Rouge, 41–106 (1971; Zbl 0289.52004)]. Let us recall that a simplicial hyperplane arrangement in the real projective space is defined via its complement, namely the irreducible components of the complement are open simplices. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones, the so-called shards.
In the paper under review, the authors provide an update on Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to \(37\) lines and their key invariants. In order to due so, the authors add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. In order to obtain their results, the authors use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. Moreover, the authors show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.