This paper deals with generalization of the Brayton-Moser network decomposition and related structural properties to a relatively large class of finite dimensional strictly causal systems, which can be described in the state-space representation form. The resulting energy-metric function is defined for dissipative systems and is induced by the output signal dissipation power. It is demonstrated that such a power-oriented approach determines both, the structure of a system representation as well as the corresponding system state space topology. A special form of physically correct internal structure of an equivalent state space representation has been derived as a natural consequence of strict causality, the state-space energy conservation, dissipativity assumption and the state minimality requirement.