Abstract We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a $2$-cocycle for connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ an affine Noetherian domain with Gelfand–Kirillov dimension $\dim (G)$, and if unipotent supported on $G$, then ${}_J\mathcal{O}(G)_{J}\co...