We study a restriction of the classification procedure for EL++ where the inference rule for complex role inclusion axioms (RIAs) is applied in a “left-linear” way in analogy with the well-known procedure for computing the transitive closure of a binary relation. We introduce a notion of left-admissibility for a set of RIAs, which specifies when a subset of RIAs can be used in a left-linear way without loosing consequences, prove a criterion which can be used to effectively check this property, and describe some preliminary experimental results analyzing when the restricted procedure can give practical improvements.