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In this paper, we study simplicial hyperplane arrangements in real projective$3$-space. We give a necessary condition for the characteristic polynomial tohave only real roots, valid also for non-simplicial arrangements. Asapplication, we obtain combinatorial inequalities which are satisfied forarrangements with splitting polynomial. This allows us to prove that there areonly finitely many different isomorphism classes of simply laced simplicialarrangements whose characteristic polynomials split over $\mathbb{R}$. We alsoprovide an updated version of a catalogue published by Gr\"unbaum and Shephardand review some conjectures of theirs.

On simplicial arrangements in $\mathbb{P}^3(\mathbb{R})$ with splitting polynomial