Summary: We improve results and proofs of our earlier paper [Appl. Algebra Eng. Commun. Comput. 28, No. 1, 31–78 (2017; Zbl 1368.13031)] and show that the ideal in question is a complete intersection. Let \(\mathbb{K}\) be a field and \(\text{M} = \mathbb{K} [x^{-1}, z^{-1}]\) denote the \(\mathbb{K}[x,z]\) submodule of Macaulay’s inverse system \(\mathbb{K}[[x^{-1},z^{-1}]]\). We regard \(z \in \mathbb{K}[x,z]\) and \(z^{-1} \in \text{M}\) as homogenising variables. An inverse form \(F\in \text{M}\) has a homogeneous annihilator ideal \(\mathcal{I}_F\) . In our earlier paper we inductively constructed an ordered pair \((f_1, f_2)\) of forms in \(\mathbb{K}[x,z]\) which generate \(\mathcal{I}_F\). We give a significantly shorter proof that accumulating all forms for \(F\) in our construction yields a minimal grlex Gröbner basis \(\mathcal{F}\) for \(\mathcal{I}_F\) (without using the theory of S polynomials or distinguishing three types of inverse forms) and we simplify the reduction of \(\mathcal{F}\). The associated Gröbner basis algorithm terminates by construction and is quadratic. Finally we show that \(f_1,f_2\) is a maximal \(\mathbb{K}[x,z]\) regular sequence for \(\mathcal{I}_F\) , so that \(\mathcal{I}_F\) is a complete intersection.