The complexity of a multivariate self-organized system results from the nonlinear interactions and feedbacks among the components. Such complexity results in a variety of attributes such as stranger attractor and 1/f fractal behavior. How does the evolutionary dynamics involving multiple variables sustain self-organization? To address this issue, we employ an information-theoretic approach to analyze the causal dynamics owing through a time series directed acyclic graph representation of a complex system. The causal dynamics quantifies the influences on the predictability of one or more lagged variables from a part or the entire historical dynamics. Specifically, we investigate how the current status of a system is influenced by (1) the entire evolutionary dynamics in the system, decomposed into an immediate and a distant parts; (2) a recent dynamics trajectory from one variable (pairwise interaction); and (3) multiple recent dynamics trajectories from several variables (multivariate interaction). We study both synthetic and observed environmental time series data. Through this analysis, consistent nonzero information from distant causal history shows the long-term dependency of the two systems, consistent with previous studies. Also, we find that self-organization is sustained by the strong influence from the immediate causal history as well as the increasing synergistic influence between self and cross-variable dependencies in distant causal history.