Lie group $SU(2)$ is the universal covering group of $SO(3)$.

I need to show that Lie group $$SU(2)$$ is the universal covering group of $$SO(3)$$ using the Adjoint representation of $$SU(2)$$.

But I am stuck at the first step of finding the adjoint representation of $$SU(2)$$. Using the adjoint representation of $$SU(2)$$ I need to produce a continuous projection $$p: SU(2)\to SO(3)$$ so that $$SU(2)$$ is the universal covering of $$SO(3)$$. Then I know a result which will imply that $$SU(2)$$ is simply connected since $$SO(3)$$ is connected.

Since $$SU(2)$$ is simple, the Killing form of $$su(2)$$is negative definite, the adjoint representation preserves the Killing form, thus its image is contained in $$SO(3)$$, it is a local diffeomorphism and since $$dim SU(2)=dim SO(3)$$ the adjoint representation is a covering (Ehresmann Lemma).
Proof of simple connectivity of $$SU(2)$$: $$SU(2)$$ is homeomorphic to the space $$S^3$$. A loop $$p:[0,1]\to S^3$$ can be contracted by:
• Ideally, picking a point not in the image of $$p$$, stereographically projecting from that point onto $$\mathbb R^3$$, and then contracting in $$\mathbb R^3$$ (which is clearly simply connected).
• To deal with paths that fill space, which blocks the above argument, exploit the uniform continuity of paths. Uniform continuity implies that the domain of $$p$$ can be broken up into intervals of size $$\delta$$ such that the image of such an interval doesn't fill every point on the sphere. Within each such interval, straighten out the path, then apply the above argument.
Proof that $$SU(2)$$ covers $$SO(3)$$: Proof 1: $$\mathfrak{su}(2)$$ is isomorphic to $$\mathfrak{so}(3)$$, while the former is simply connected. (Is this correct?) Proof 2: Interpret $$\mathfrak{su}(2)$$ as a set of matrices (which is justified by the fact that $$SU(2)$$ is a matrix group) and a vector space too. Let $$u$$ and $$v$$ be in $$\mathfrak{su}(2)$$; observe that $$e^{v\theta/2}\,u\,e^{-v\theta/2}$$ is a rotation of $$u$$. The mapping $$u \in \mathfrak{su}(2) \mapsto e^{v\theta/2}\,u\,e^{-v\theta/2}$$ is the adjoint representation of $$e^{v\theta/2}$$, and every element of $$SU(2)$$ can be expressed as $$e^{v\theta/2}$$.