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