The conditional autoregressive model is a routinely used statistical model for areal data that arise from, for instances, epidemiological, socio-economic or ecological studies. Various multivariate conditional autoregressive models have also been extensively studied in the literature and it has been shown that extending from the univariate case to the multivariate case is not trivial. In this paper, we approach the multivariate modeling from an element-based perspective instead of the traditional vector-based perspective. We focus on the joint adjacency structure of elements and discuss graphical structures for both the spatial and non-spatial domains. We assume that the graph for the spatial domain is generally known and fixed while the graph for the non-spatial domain can be unknown and random. We propose a very general specification for the multivariate conditional modeling and then focus on three special cases, which are linked to well known models in the literature.