The Thue–Morse continued fractions in characteristic 2 are algebraic

Let $a, b$ be distinct, non-constant polynomials in $\mathbb F_2 [z]$. Let $\xi_{a, b}$ be the power series in $\mathbb F_2((z^{-1}))$ whose sequence of partial quotients is the Thue–Morse sequence over $\{a, b\}$. We establish that $\xi_{a,b}$ is algebraic of degree $4$.