On the Cox ring of blowing up the diagonal

We compute the Cox rings of the blow-ups ${\mathrm{B}\mathrm{l}}_{\mathrm{\Delta }}(X^{\mathrm{\prime }}\times X^{\mathrm{\prime }})$ and ${\mathrm{B}\mathrm{l}}_{\mathrm{\Delta }}({\mathbb{P}}_1^n)$ where $X^{\mathrm{\prime }}$ is a product of projective spaces and $\mathrm{\Delta }$ is the (generalised) diagonal.