= Solution
In the printed phased <Bell basis>, the actions on its four kets are
$$
\begin{array}{c|ccc}
&\sigma_A^x&\sigma_A^y&\sigma_A^z\\\hline
|1\rangle&i|2\rangle&-i|3\rangle&i|4\rangle\\
|2\rangle&-i|1\rangle&-i|4\rangle&-i|3\rangle\\
|3\rangle&-i|4\rangle&i|1\rangle&i|2\rangle\\
|4\rangle&i|3\rangle&i|2\rangle&-i|1\rangle
\end{array}
$$
Each <matrix> is purely imaginary and antisymmetric. Let $T$ have these four basis kets as columns in the computational basis. The <infinitesimal generators> of the standard rotation $U_A=\exp[-i\theta\,\mathbf n\cdot\boldsymbol\sigma_A/2]$ are
$$
G_a=-\frac i2T^\dagger(\sigma_a\otimes I)T.
$$
Using the displayed actions gives the phase-convention-specific identities
$$
\boxed{G_x=-L_x^-,\qquad G_y=L_y^-,\qquad G_z=-L_z^-.}
$$
They are real <antisymmetric matrices>. A finite $SU(2)$ rotation on $A$ therefore acts on the magic-basis coefficient vector as $b\mapsto Ob$, with $O\in SO(4)$. Consequently
$$
q'=(Ob)^{\mathsf T}(Ob)=b^{\mathsf T}O^{\mathsf T}Ob=b^{\mathsf T}b=q.
$$
It preserves <concurrence> and hence \b[preserves the reduced <eigenvalues> and <entanglement entropy>]. A general $U(2)$ operation adds a common <phase> to $b$; that changes $q$ by a <phase> of twice the angle but leaves $|q|$ unchanged. The real-rotation description of local transformations is a characteristic property of the <magic basis>.<Entanglement of a Pair of Quantum Bits by Scott Hill and William K. Wootters>[After Eq.(6), real magic-basis matrices and independent local transformations]{ref}
As an independent direct check, a local unitary sends $\rho_A$ to $U_A\rho_AU_A^\dagger$, which is a unitary similarity transformation and cannot change its <spectrum>. This <local-unitary invariance of entanglement entropy> also explains why merely rotating one subsystem cannot create <entanglement>.
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