  We show that bounded cohomology stabilizes along sequences of classical Lie
groups, and along sequences of lattices in them. Our method is based on a
criterion from (arXiv:2307.12808) which adapts Quillen's stability method to
the setting of bounded cohomology. This criterion is then applied to a family
of measured complexes, the so-called Stiefel complexes, associated to any
vector space endowed with a non-degenerate sesquilinear form.
