A two-dimensional rationality problem and intersections of two quadrics

Let k be a field with char $$k\ne 2$$ and k be not algebraically closed. Let $$a\in k{\setminus } k^2$$ and $$L=k(\sqrt{a})(x,y)$$ be a field extension of k where x, y are algebraically independent over k. Assume that $$\sigma $$ is a k-automorphism on L defined by $$\begin{aligned} \sigma : \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c\big (x+\frac{b}{x}\big )+d}{y} \end{aligned}$$ where $$b,c,d \in k$$ , $$b\ne 0$$ and at least one of c, d is non-zero. Let $$L^{\langle \sigma \rangle }=\{u\in L:\sigma (u)=u\}$$ be the fixed subfield of L. We show that $$L^{\langle \sigma \rangle }$$ is isomorphic to the function field of a certain surface in $$\mathbb {P}^4_k$$ which is given as the intersection of two quadrics. We give criteria for the k-rationality of $$L^{\langle \sigma \rangle }$$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Thelene.