Is there a version of this written somewhere for graded commutative quadratic algebras? Here I mean algebras with a presentation $$A = S(V) / (R)$$ where $$V$$ is a graded vector space (let's say positively graded), $$S(V)$$ is the free graded commutative algebra on $$V$$, and $$R \subset S^{(2)}(V)$$ is a set of quadratic relations?
I guess one just needs to introduce relations of the form $$x_i x_j = \pm x_j x_i$$ instead of simply $$x_i x_j = x_j x_i$$. Theorem 8.1 in Chapter 4 of Quadratic Algebras already allows for general relations of the form $$x_i x_j = q_{ij} x_j x_i$$ for scalars $$q_{ij}$$, and in Section 10 it is proved that a $$\mathbb{Z}$$-algebra (generalization of graded associative algebras) which admits a PBW basis is Koszul. But it'd be nice to have something precise to reference...