Overlooked restrictions on Euler angles in quantum computation

Let $X,Y,Z$ denote the Pauli matrices. For $\vec{n}=(n_x,n_y,n_z) \in {\bf R}^3$ with $n_x^2+n_y^2+n_z^2=1$ and $\theta\in{\bf R}$, put $R_{\vec{n}}(\theta)=\cos(\theta/2)I-i\sin(\theta/2)(n_x X+n_yY+n_z Z)$. Put $R_{y}(\theta)=R_{(0,1,0)}(\theta)$ and $R_{z}(\theta)=R_{(0,0,1)}(\theta)$. Theorem: Assume $\alpha,\gamma,\theta\in{\bf R}$, $\vec{n}=(n_x,n_y,n_z)\in{\bf R}^3$ and $n_x^2+n_y^2+n_z^2=1$. Then, there exists some $\beta,\delta\in{\bf R}$ satisfying $R_{\vec{n}}(\theta)=e^{i\alpha}R_{z}(\beta)R_{y}(\gamma)R_{z}(\delta)$ if and only if (iff) $e^{i\alpha}=1$ or $-1$, and $\sqrt{1-n_z^2}|\sin(\theta/2)|=|\sin(\gamma/2)|$. Corollary: Assume $\alpha,\gamma\in{\bf R}$, $\vec{n}=(n_x,n_y,n_z)\in{\bf R}^3$ and $n_x^2+n_y^2+n_z^2=1$. Then, there exist some $\beta,\delta,\theta\in{\bf R}$ such that $e^{i\alpha}R_{z}(\beta)R_{\vec{n}}(\theta)R_{z}(\delta)=R_{y}(\gamma)$ iff $e^{i\alpha}=1$ or $-1$, and $|\cos(\gamma/2)|\ge|n_z|$. This corollary shows a widespread fallacy on universal gates in quantum computation. Namely, when $|\cos(\gamma/2)|<|n_z|<1$, according to a claim often found in textbooks, $R_{y}(\gamma)$ could be written as $e^{i\alpha}R_{z}(\beta)R_{\vec{n}}(\theta)R_{z}(\delta)$ for some $\alpha,\beta,\delta,\theta\in{\bf R}$. This is untrue by the corollary.