If 1, u, ū, g, ug are free and u·R(u)·g = N(u) with R ≠ 0, deg R ≤ 2, deg N ≤ 3, their span contains u², ū² or some 1/(u − a)

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$ with $\rho = u\bar u$ algebraic (so $\bar u = \rho/u$). Let $g \in \mathbb{C}$ with $1, u, \bar u, g, ug$ free over $\overline{\mathbb{Q}}$, and suppose $u\,R(u)\,g = N(u)$ for some $R, N \in \overline{\mathbb{Q}}[X]$ with $R \neq 0$, $\deg R \leq 2$ and $\deg N \leq 3$. Then $W_g = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, g, ug)$ contains $u^2$, $\bar u^2$ or some $1/(u - a)$ with $a \in \overline{\mathbb{Q}}^\times$.

Proof. Put $D = uR(u) \neq 0$ and $\varphi(M) = M(u)/D$, a $\overline{\mathbb{Q}}$-linear map on $\overline{\mathbb{Q}}[X]$. The generators $1, u, \bar u, g, ug$ are $\varphi(XR), \varphi(X^2R), \varphi(\rho R), \varphi(N), \varphi(XN)$, all of degree at most $4$, so $W_g$ lies in the image of the polynomials of degree below $5$, of dimension at most $5$. As $\dim W_g = 5$, the two are equal. If $\deg R \leq 1$ then $u^2 = \varphi(X^3R)$; if $R(0) = 0$, $R = XR_1$ and $\bar u^2 = \varphi(\rho^2R_1)$; otherwise $R$ has a root $b \in \overline{\mathbb{Q}}^\times$, $R = (X - b)R_1$ and $1/(u - b) = \varphi(XR_1)$.

Novelty. Not found in the sources read; routine.