  In a recent article, Cumplido and Paris studied the question of
commensurability between Artin groups of spherical type. Their analysis left
six cases undecided, for the following pairs of Artin groups: $(F_4,D_4)$,
$(H_4,D_4)$, $(F_4,H_4)$, $(E_6,D_6)$, $(E_7,D_7)$, and $(E_8,D_8)$. In this
note we resolve the first two of these cases, namely, we show that the Artin
groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$.
As a key step, we realize the abstract commensurator of the Artin group of type
$D_4$ as the extended mapping class group of the torus with three punctures. We
also find the automorphism group of the Artin group of type $D_4$ and obtain a
description of torsion elements, their orders and conjugacy classes in all
irreducible Artin groups of spherical type modulo their centers.
