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The concept of informational independence plays a key role in most knowledge-based systems. J. Pearl and his co-researchers have analysed the basic properties of the concept and have formulated an axiomatic system for informational independence. This axiomatic system focuses on independences among mutually disjoint sets of variables. We show that in the context of probabilistic independence a focus on disjoint sets of variables can hide various interesting properties. To capture these properties, we enhance Pearl's axiomatic system with two additional axioms. We investigate the set of models of the thus enhanced system and show that it provides a better characterisation of the concept of probabilistic independence than Pearl's system does. In addition, we observe that both Pearl's axiomatic system and our enhanced system o er inference rules for deriving new independences from an initial set of independence statements and as such allow for a normal form for representing independence. We address the normal forms ensuing from the two axiomatic systems for informational independence.