Aftershock cascades are a characteristic feature of natural seismicity, but underlying mechanisms remain debated. Here, we experimentally explore the presence or absence of aftershocks during failure of intact rock and slip on newly created laboratory faults. We show that the overall activity increase and spatial localization of acoustic emission (AE) events during fracture nucleation occurs without temporal (Omori-type) correlations. Our analysis shows that this absence of aftershock sequences occurs even beyond peak stress and also when a macroscopic fracture has formed post peak-stress and propagates. Instead, aftershock triggering does occur during post-fracture stress relaxation along the newly created lab-fault and in the presence of large-scale stress heterogeneities, for example, imposed by a saw-cut notch. The detected aftershocks in these cases can be described by standard seismological relationships such as a modified Omori-Utsu relation and its associated inter-event time distribution and productivity relation. Moreover, AE within all experiments follow the Gutenberg-Richter relation, with smaller (Formula presented.) -values for triggered events compared to non-triggered events. Performing full-moment tensor inversions, we find that seismic events with significant isotropic, compaction components play an important role for aftershock triggering. The resulting triggered events tend to have focal mechanisms similar to their trigger. Seismic events with predominant tensile components, on the other hand, show little evidence for aftershock triggering. This opens up a new perspective on aftershocks, going beyond not only the rate-and-state paradigm limited to purely frictional sliding or shear events but also the mainshock attribute paradigm where mainshock attributes control aftershock patterns.