Unpacking a product-form independence hypothesis

Suppose E, F and G are 3 events such that $P(E\cap F\cap G)=P(E)\cdot P(F)\cdot P(G)$ and $P(E\cap F\cap G)=P(E)\cdot P(F\cap G)$. Consider the statements: (I) F and G are independent events. (II) E and $F^C\cup G^C$ are independent events. (III) $E^C$ and $F\cap G$ are independent events. (IV) E, F and G are mutually independent events.