Commonsense knowledge (CSK) about concepts and their properties is useful for AI applications. Prior works like ConceptNet, COMET and others compiled large CSK collections, but are restricted in their expressiveness to subject-predicate-object (SPO) triples with simple concepts for S and strings for P and O. This paper presents a method, called ASCENT++, to automatically build a large-scale knowledge base (KB) of CSK assertions, with refined expressiveness and both better precision and recall than prior works. ASCENT++ goes beyond SPO triples by capturing composite concepts with subgroups and aspects, and by refining assertions with semantic facets. The latter is important to express the temporal and spatial validity of assertions and further qualifiers. ASCENT++ combines open information extraction with judicious cleaning and ranking by typicality and saliency scores. For high coverage, our method taps into the large-scale crawl C4 with broad web contents. The evaluation with human judgements shows the superior quality of the ASCENT++ KB, and an extrinsic evaluation for QA-support tasks underlines the benefits of ASCENT++. A web interface, data and code can be accessed at https://www.mpi-inf.mpg.de/ascentpp.