= Solution
In the defining representation,
$$
H=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad
E_+=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad
E_-=\begin{pmatrix}0&0\\1&0\end{pmatrix}.
$$
Multiplying the three factors in the <disentangling identity> and comparing with $e^{i\theta\sigma_1/2}$ gives
$$
\alpha=\gamma=\tan\frac\theta2,
\qquad
\exp\!\left(\frac{i\beta H}{2}\right)
=\begin{pmatrix}\cos(\theta/2)&0\\0&\sec(\theta/2)\end{pmatrix}.
$$
This factorization is the coordinate patch where $\cos(\theta/2)\ne0$.
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