Reading the Nielsen and Chuang, I saw that every unitary operator $U$ can be written as $e^{i\alpha} R_n(\theta)$ for some well chosen $n \in \mathbb{R}^3$ and $0 \leq \theta < 2\pi$.
I would like to know if there is a simple criterion to ensure that $\alpha=0$, without knowing the value of $n$ or $\theta$.
As a consequence of it, using the fact that $e^{i\alpha}$ "disappear" when $U$ is seen as a rotation of the Bloch Sphere, and the fact that the composition of two rotations on $\mathbb{R}^3$ is also a rotation, I will have the proof that the composition of two morphisms $R_{n_1}(\theta_1) R_{n_2}(\theta_2)$ is on the form $R_{n_3}(\theta_3)$.
This problem is thus linked to the fact that the Bloch Sphere identify the unitary transformations that are equal up to a global phase (if I am right : $U_1$ and $U_2$, seen as morphisms on the Bloch sphere, are equal iff there are equal up to a global phase).
Edit: I precise I am interested by the result even if the global phase can be removed in one-qubit gates. One of the reasons is because of the multi-qubits gates, such as for exemple the two-bits gate that is a controled one-qubit gate.