$4$ planes in ${\mathbb{R}}^4$

We establish a homeomorphism between the moduli space $A_{4,k}^{\rm ord}(\mathbb{R})$ of ordered $k$-tuples $(H_1,\ldots ,H_k)$ of 2-dimensional linear subspaces $H_i \subset \mathbb{R}^4$ (mod ${\rm GL}_4(\mathbb{R})$) and the quotient by simultaneous conjugation of a certain open subset $({\rm GL}_2^{k-3})^* \subset ({\rm GL}_2(\mathbb{R}))^{k-3}$. For $k=4$, this leads to an explicit computation of the moduli space $A_{4,4}(\mathbb{R})$ of central 2-arrangements in $\mathbb{R}^4$ mod ${\rm GL}_4(\mathbb{R})$ and its subspace $A_{2,4}({\mathbb{C}})$ of those classes that contain a complex hyperplane arrangement.