Consider an algebraic irrational number in $(0,1)$ with binary expansion $x = \sum_{n\ge 1} \frac{a_n}{2^n}$. Is it possible that the number $\sum_{n\ge 1}\frac{a_{2n}}{2^n}$ is again algebraic irrational ?

My opinion is no. Heuristics: it should be difficult to determine binary expansions of other algebraic irrationals, even if we have a supply of other expansions, except for trivial operations. My previous question (see link) is of a similar flavor.