Pythagoras numbers of orders in biquadratic fields

We examine the Pythagoras number $\mathcal{P}({\mathcal{O}}_K)$ of the ring of integers ${\mathcal{O}}_K$ in a totally real biquadratic number field $K$. We show that the known upper bound $7$ is attained in a large and natural infinite family of such fields. In contrast, for almost all fields $$\mathbb{Q}$(\sqrt5, \sqrt{s})$weprove$$\mathcal{P}$($\mathcal{O}$_K)=5$\mathrm{.}Furtherweshowthat$5$ is a lower bound for all but seven fields $K$ and $6$ is a lower bound in an asymptotic sense.