Can these seven be reduced to three basic rules? - controlling for a node increases correlation among pairs where both are ancestors - controlling for a node

Can these seven be reduced to three basic rules? - controlling for a node increases correlation among pairs where both are ancestors - controlling for a node does not affect (the lack of) correlation among pairs where at least one is categorically unrelated (shares no ancestry with that node) - controlling for a node decreases correlation among pairs where both are related but at least one is not an ancestor