Simple Properties in SHOIN(D) are Properties without transitive Subproperties, while in SHROIQ(D) it is a little bit more complicated because General Property Inclusion has to be considered. There simple Properties are Properties that are not on the right side of the Property Inclusion or that are the inverse of other simple Properties, or they are only on the right side on the Property inclusion where on the left site is a simple Property.
SHROIQ(D) general Property Inclusion has to be considered
are all Properties
Several Expressions are only permitted for simple Properties, otherwise the entire Knowledge Base would become undecidable. For example only for simple properties one can use the Qualified Number Restrictions, one can not use them on complex properties that are based on General Role Inclusions. On the other hand irreflexive and Disjunctive Properties are also only permitted for simple Properties. In the same way if one uses the Existential Quantifier or one uses the complement of an Instantiation, it is only allowed for simple Properties and not for complex Properties, because if one would apply this Properties on Properties that depend on General Role Inclusion then the Knowledge Base and the Logic would become undecidable.