Do Bloch sphere rotations span $SU(2)$ (up to a global phase)?

Question

It is well known that the Pauli group $\{I,X,Y,Z\}$ spans the group of $2\times2$ unitary matrices, $SU(2)$, for example see this link.

A general Bloch sphere rotation by an angle $\alpha$ about an arbitrary axis, $\hat{n}$ is given by:

$$R_{\hat{n}}\left(\alpha\right)=\cos\left(\frac{\alpha}{2}\right)I-i\sin\left(\frac{\alpha}{2}\right)\left(n_x X+n_y Y+n_z Z\right)$$

Is it true, that up to a global phase, Bloch sphere rotations span $SU(2)$?

If so, can it be proved without using group theory and abstract algebra?

actually technically the Pauli group has $8$ elements: $\pm \mathbb{I},\pm X,\pm Y,\pm Z$. See https://doi.org/10.1063/1.528006 — ZeroTheHero, Jul 21 '22 at 21:45
All you need to do is to show that any SU(2) matrix has the form $R_{\hat n}$ so write the most general SU(2) matrix and find the coefficients $n_i$ and $\alpha$ using trace orthogonality of the Pauli matrices… — ZeroTheHero, Jul 21 '22 at 21:48
$SU(2)$ is not a vector space, so it isn't the span of anything. The four matrices you've given span all of $\mathbb C^{2\times2},$ which includes $SU(2)$ but also a lot of other junk. $\mathfrak{su}(2)$ is a vector space and is spanned by just $X,Y,Z.$ The map $\exp:\mathfrak{su}(2)\to SU(2)$ surjects, and your $R$ should be explainable as interpreting $\hat n,a$ as a element of $\mathfrak{su}(2)$ and then exponentiating. — HTNW, Jul 21 '22 at 21:56
Related : (1) Understanding the Bloch sphere. (2) Why is there this relationship between quaternions and Pauli matrices?. — Frobenius, Jul 22 '22 at 03:29
Another possible duplicate: https://physics.stackexchange.com/q/174562/2451 — Qmechanic, Jul 22 '22 at 06:56

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