 Solution

Let $\beta =a+b\alpha \in {\mathcal{O}}_L$. The nontrivial $K$-automorphism sends $\alpha$ to $-\alpha$, soAlso $\alpha$ is integral, hence $\alpha \beta$ is integral, andBut $4+\sqrt{17}$ has norm $-1$, so it is a unit of ${\mathcal{O}}_K$. Multiplying by its inverse proves $2b\in {\mathcal{O}}_K$.