Suppose $\hat{m},\hat{n}$ are non-parallel real unit vectors in 3D. Show that an arbitrary single qubit unitary operator $U$ may be written as $$ U=e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\hat{n}}(\delta) $$ for appropriate choices of $\alpha,\eta,\gamma,\delta$, where $R_{\hat{n}}(\theta)=\exp\Big(\frac{-i\theta}{2}\hat{n}.\vec{\sigma}\Big)=\cos(\theta/2)I-i(\hat{n}.\vec{\sigma})\sin(\theta/2)$, $\vec{\sigma}=(X,Y,Z)$, $X=\begin{bmatrix}0&1\\1&0\end{bmatrix},Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix},Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ are Pauli matrices and $\hat{n}$ is some real 3D unit vector.
We can easily prove that $U=e^{i\alpha}R_{z}(\beta)R_{y}(\gamma)R_{z}(\delta)$ where $R_{x}(\theta)=\exp\Big(\frac{-i\theta}{2}X\Big)=\cos(\theta/2)I-i\sin(\theta/2)X=\begin{bmatrix}\cos\theta/2&-i\sin\theta/2\\-i\sin\theta/2&\cos\theta/2\end{bmatrix},R_{y}(\theta)=\exp\Big(\frac{-i\theta}{2}Y\Big)=\cos(\theta/2)I-i\sin(\theta/2)Y=\begin{bmatrix}\cos\theta/2&-\sin\theta/2\\\sin\theta/2&\cos\theta/2\end{bmatrix},R_{z}(\theta)=\exp\Big(\frac{-i\theta}{2}Z\Big)=\cos(\theta/2)I-i\sin(\theta/2)Z=\begin{bmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{bmatrix}$
My reference asks to prove $U=e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\hat{n}}(\delta)$ from $U=e^{i\alpha'}R_{z}(\beta')R_{y}(\gamma')R_{z}(\delta')$ ?
Note: Please check Page 176-Quantum Computation and Quantum Information Nielsen & Chuang
Attempt
Thanks @BenGrossmann for the suggestion,
Since $R_{\hat{m}}(\theta)$ rotates a Bloch vector into any other Bloch vector which includes all possible qubit states upto a global phase factor, an arbitrary unitary operator can be written in the form $U=e^{i\alpha}R_{\hat{m}}(\theta)$
\begin{align} U'=e^{ia}R_{\hat{m}}(t)=e^{ia}R_{z}(b)R_{y}(c)R_{z}(d)\\ \implies R_{y}(c)=(R_{z}(b))^{-1}R_{\hat{m}}(t)(R_{z}(d))^{-1}\\ \boxed{R_{y}(c)=R_{z}(-b)R_{\hat{m}}(t)R_{z}(-d)}\tag{Eq.1}\\ \end{align}
It is easy to prove $U=e^{i\alpha}R_{z}(b')R_{y}(c')R_{z}(d')$ for real parameters $\alpha',b',c',d'$ $$ U=e^{i\alpha}R_{z}(b')R_{y}(c')R_{z}(d')=e^{i\alpha}R_{z}(b').R_{z}(-b)R_{\hat{m}}(t)R_{z}(-d).R_{z}(d')\\ =e^{i(\alpha)}R_{z}(b'-b)R_{\hat{m}}(t)R_{z}(d'-d)=e^{i(\alpha)}R_{z}(\beta)R_{\hat{m}}(\gamma)R_{z}(\delta)\\ $$ \begin{align} \implies \boxed{U=e^{i(\alpha)}R_{\hat{m}}(\phi)=e^{i(\alpha)}R_{z}(\beta)R_{\hat{m}}(\gamma)R_{z}(\delta)=e^{i(\alpha')}R_{z}(\beta')R_{y}(\gamma')R_{z}(\delta')}\tag{Eq.2} \end{align}
But, how do use this expression $U=e^{i(\alpha)}R_{z}(\beta)R_{\hat{m}}(\gamma)R_{z}(\delta)$ to reach $U=e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\hat{n}}(\delta)$ ?
My Attempt
$U=e^{ik}R_{\hat{n}}(t)=e^{iq}R_{z}(b)R_{y}(c)R_{z}(d)\implies R_{\hat{n}}(t)=e^{i(q-k)}R_{z}(b)R_{y}(c)R_{z}(d)=e^{i a}R_{z}(b)R_{y}(c)R_{z}(d)$
$R_{\hat{n}}(\beta)=e^{i a}R_{z}(b)R_{y}(c)R_{z}(d)$ and $R_{\hat{n}}(\delta)=e^{i a'}R_{z}(b')R_{y}(c')R_{z}(d')$ and $R_{\hat{m}}(\gamma)=e^{i h}R_{z}(e)R_{y}(f)R_{z}(g)$.
$$ e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\hat{n}}(\delta)=e^{i\alpha}e^{i a}R_{z}(b)R_{y}(c)R_{z}(d).R_{\hat{m}}(\gamma).e^{i a'}R_{z}(b')R_{y}(c')R_{z}(d')\\ =e^{i (\alpha+a+a')}R_{z}(b)R_{y}(c)R_{z}(d).R_{\hat{m}}(\gamma).R_{z}(b')R_{y}(c')R_{z}(d')\\ =e^{i (\alpha+a+a')}R_{z}(b).\big[R_{y}(c)R_{z}(d)R_{\hat{m}}(\gamma)R_{z}(b')R_{y}(c')\big].R_{z}(d')\\ =e^{i (\alpha')}R_{z}(b).\big[R_{y}(c)R_{z}(d)R_{\hat{m}}(\gamma)R_{z}(b')R_{y}(c')\big].R_{z}(d') $$ From Eq.2, $R_{y}(c)R_{z}(d)R_{\hat{m}}(\gamma)R_{z}(b')R_{y}(c')=R_{y}(c)R_{\hat{n}}(\theta)R_{y}(c')$, which corresponds to rotating the frame about y-axis by $c'$, then rotate about the unit vector $\hat{n}$ by $\theta$ and rotate the frame about y axis by c, i.e., is some rotation about a unit vector $\hat{N}$ by some angle $\epsilon$
$\therefore$ $R_{y}(c)R_{z}(d)R_{\hat{m}}(\gamma)R_{z}(b')R_{y}(c')=R_{y}(c)R_{\hat{n}}(\theta)R_{y}(c')=R_{\hat{N}}(\epsilon)$ $$ e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\hat{n}}(\delta)=e^{i (\alpha')}R_{z}(b).R_{\hat{N}}(\epsilon).R_{z}(d')\\ =e^{i (\alpha')}R_{\hat{r}}(\eta)=U $$
i.e., Any unitary matrix $U=e^{i (\alpha')}R_{\hat{r}}(\eta)$ can be written as $e^{i\alpha}R_{\hat{n}}(\beta)R_{\hat{m}}(\gamma)R_{\hat{n}}(\delta)$ for some real numbers $\alpha,\beta,\gamma,\delta$.