Satisfiability (SAT) solvers often benefit from clauses learned by the DPLL procedure, even though they are by definition redundant. In addition to those derived from conflicts, the clauses learned by dominator analysis during the deduction procedure tend to produce smaller implication graphs and sometimes increase the deductive power of the input CNF formula. We extend dominator analysis with an efficient self-subsumption check. We also show how the information collected by dominator analysis can be used to detect redundancies in the satisfied clauses and, more importantly, how it can be used to produce supplemental conflict clauses. We characterize these transformations in terms of deductive power and proof conciseness. Experiments show that the main advantage of dominator analysis and its extensions lies in improving proof conciseness.