Ontology-based data access consists in using ontologies while querying data. Due to the high complexity of this problem, considering lightweight description logics like EL is especially relevant. Another strand of research is based on existential rules. In this paper, we use this latter formalism in order to cover EL with the same complexity of reasoning while allowing any predicate arity and some cycles on variables. We then add complex role inclusions to enhance expressivity, while staying polynomial in data complexity and generalizing existing results. In particular, we consider transitivity and right/left identity rules, which do not behave well with respect to usual decidability paradigms.