The span of Theorem E's involution class, span(1, u, ū, v̄² + e v², β v̄³ + ζ v² + e v³) with v = u/s, is Theorem F's cyclic space D2b for the pencil (s X, X² + β s²)

Let $u \in \mathbb{C}$ be transcendental, that is, $u \notin \overline{\mathbb{Q}}$ (the algebraic numbers); spans and linear independence are over $\overline{\mathbb{Q}}$. Let $s \in \overline{\mathbb{Q}}$ with $s^2 = u\bar u$, put $v = u/s$, and let $e, \beta, \zeta \in \overline{\mathbb{Q}}$ with $\beta \neq 0$. Then there are $P', Q', S_1, S_2 \in \overline{\mathbb{Q}}[X]$ with $P'(u) = su$, $Q'(u) = u^2 + \beta s^2$, $S_1(u) = s/\beta + (e/s)u^2$ and $S_2(u) = (\zeta/s)u^2$, such that $P', Q'$ are coprime, $\max(\deg P', \deg Q') = 2$, $\deg S_1, \deg S_2 \leq 2$, and $\mathrm{span}(1, u, \bar u, \bar v^2 + ev^2, \beta\bar v^3 + \zeta v^2 + ev^3) = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), Q'(u)(Q'(u)S_1(u) + P'(u)S_2(u))/(uP'(u)^2))$.

The left side is the space of Theorem E's involution class (Q), which asks in addition $e\bar e = 1$, $\beta\bar\beta = 1$ and $\zeta = -\beta e^2\bar\zeta$; the identity needs none of these, and no dimension hypothesis. So every hull of class (Q) has Theorem F's shape (D2b), with $P' = sX$ and $Q' = X^2 + \beta s^2$.

Proof. As $u \neq 0$, $s \neq 0$; and $s$ is real, since $s^2 = u\bar u = |u|^2 > 0$. So $\bar u = s^2/u$ and $\bar v = s/u$. Take $P' = sX$ and $Q' = X^2 + \beta s^2$, coprime by the Bézout identity $-(\beta s^3)^{-1}X\cdot P' + (\beta s^2)^{-1}Q' = 1$, with $\max(\deg P', \deg Q') = 2$; and $S_1 = s/\beta + (e/s)X^2$, $S_2 = (\zeta/s)X^2$. Write $n_0 = \bar v^2 + ev^2$, $n_1 = \beta\bar v^3 + \zeta v^2 + ev^3$, $g_1 = Q'(u)S_1(u)/(uP'(u))$ and $g_2 = Q'(u)(Q'(u)S_1(u) + P'(u)S_2(u))/(uP'(u)^2)$. Two exact identities hold: $g_1 - n_0 = \beta e + \beta^{-1}$ and $g_2 - n_1 = \beta\zeta + ((2\beta e + \beta^{-1})/s)\,u + ((\beta^2e + 2)/s)\,\bar u$. With $\bar u = s^2u^{-1}$ and $u^{-1} = s^{-2}\bar u$, each generator of either span is an explicit $\overline{\mathbb{Q}}$-combination of the generators of the other, and the two inclusions are checked by field arithmetic. No dimension or independence argument is used, and none of the constraints $e\bar e = \beta\bar\beta = 1$, $\zeta = -\beta e^2\bar\zeta$ of class (Q).

Novelty. Not found in the sources read; routine. An identity of spans that writes Theorem E's involution class (Q) in the coordinates of Theorem F: every hull of class (Q) has shape (D2b), with the pencil $(sX, X^2 + \beta s^2)$, whose ratio $Q'(u)/P'(u) = (u + \beta\bar u)/s$ is that of the configuration of class (Q). That configuration is in print: the proof of Diaz (2004, Théorème 2), at $x = (1, (u + \beta\bar u)/s)$; also Diaz (2007, Théorème 4(1)).