  K-theoretic Hall algebras (KHAs) of quivers with potential $(Q,W)$ are a
generalization of preprojective KHAs of quivers, which are conjecturally
positive parts of the Okounkov-Smironov quantum affine algebras. In particular,
preprojective KHAs are expected to satisfy a Poincar\'e-Birkhoff-Witt theorem.
We construct semi-orthogonal decompositions of categorical Hall algebras using
techniques developed by Halpern-Leistner, Ballard-Favero-Katzarkov, and
\v{S}penko-Van den Bergh. For a quotient of $\text{KHA}(Q,W)_{\mathbb{Q}}$, we
refine these decompositions and prove a PBW-type theorem for it. The spaces of
generators of $\text{KHA}(Q,0)_{\mathbb{Q}}$ are given by (a version of)
intersection K-theory of coarse moduli spaces of representations of $Q$.
