 Solution

Identify ${\mathbb{R}}^3$ with the traceless Hermitian matrices $X=x\cdot \sigma$, where ${\sigma }_i$ are the Pauli matrices. Then $\det X=-|x{|}^2$. For $U\in SU(2)$ the action $X\mapsto UX{U}^{†}$ preserves trace, Hermiticity and determinant, so gives an orthogonal transformation of ${\mathbb{R}}^3$. Since $SU(2)$ is connected, the determinant of this transformation is $+1$.

An element in the kernel commutes with every ${\sigma }_i$, hence is scalar. Unitarity and determinant one restrict it to $\pm I$. Every three-dimensional rotation is a rotation through some angle $\theta$ around a unit axis $n$. It is obtained byIndeed multiplication with ${\sigma }_i{\sigma }_j={\delta }_{ij}I+i{ϵ}_{ijk}{\sigma }_k$ gives the Rodrigues rotation formula for $UX{U}^{†}$. This proves surjectivity, not just equality of Lie-algebra dimensions. ThereforeA full $2\pi$ rotation lifts from $I$ to $-I$, explaining the double cover and the Spin group interpretation.