  We determine the exact structure and action of Levi subgroups of parabolic
subgroups of groups of Lie type related to thick, irreducible, spherical
buildings of simply laced type. Therefore we introduce the special and general
projectivity groups attached to simplices $F$. If the residue of $F$ is
irreducible, we determine the permutation group of the projectivity groups of
$F$ acting on the residue of $F$ and show that this determines the precise
action of the Levi subgroup of a parabolic subgroup on the corresponding
residue. This reveals three special cases for the exceptional types $E_6$,
$E_7$, $E_8$. Furthermore, we establish a general diagrammatic rule to decide
when exactly the special and general projectivity groups of $F$ coincide.
