The Dirac belt trick: a 2Ï€ rotation is non-trivial, a 4Ï€ rotation is trivial

Idea. Whether a loop is trivial is decided by whether its lift to the universal cover $S^3$ is closed.

Lifting $\gamma$. By the previous computation, $\Phi\bigl(\cos\frac\theta2+k\sin\frac\theta2\bigr)$ is the rotation by $\theta$ about the third axis ($u=k$), that is $R_\theta\oplus1=\gamma(\theta)$. So
$$\tilde\gamma(\theta)=\cos\frac\theta2+k\sin\frac\theta2,\qquad\theta\in[0,2\pi],$$
is the lift of $\gamma$ starting at $\tilde\gamma(0)=1$. It ends at $\tilde\gamma(2\pi)=\cos\pi+k\sin\pi=-1\ne1$: the lift is not closed.

$\gamma$ is not null-homotopic. Suppose $H:[0,1]\times[0,2\pi]\to SO(3)$ were a homotopy of loops based at $I$ from $\gamma$ to the constant loop. By the homotopy lifting property of coverings, $H$ lifts to $\tilde H$ with $\tilde H(0,\cdot)=\tilde\gamma$ and $\tilde H(s,0)=1$ for all $s$ (the path $s\mapsto H(s,0)=I$ lifts to the constant path $1$), and the endpoint $\tilde H(s,2\pi)$ lies in the discrete fibre $\{\pm1\}$ above $I$ and varies continuously in $s$, so it is constant. But for the constant loop the lift is constant $1$ with endpoint $1$, whereas the endpoint of $\tilde\gamma$ is $-1$. Contradiction. So $\gamma$ is not null-homotopic.

$\gamma^2$ is null-homotopic. The loop $\gamma^2(\theta)=R_\theta\oplus1$, $\theta\in[0,4\pi]$, lifts to $\tilde\gamma^{(2)}(\theta)=\cos\frac\theta2+k\sin\frac\theta2$, $\theta\in[0,4\pi]$, which ends at $\cos2\pi+k\sin2\pi=1$: a closed loop in $S^3$. Since $S^3$ is simply connected, it contracts to the constant loop within $S^3$ via a homotopy $\tilde H$; composing with $\Phi$ gives a homotopy of $\gamma^2$ to the constant loop in $SO(3)$ (the base point stays fixed).

Physical interpretation (Dirac's belt). Rotating a body (with a belt attached to a fixed wall) by $2\pi$ twists the belt in a way that cannot be undone without rotating the body back, whereas after a rotation by $4\pi$ the belt can be untangled by moving it around the body, without rotating the body. This models spin-$\frac12$ particles: their wave function changes sign under a $2\pi$ rotation.

Remark. Equivalently, $[\gamma]$ is the generator of $\pi_1(SO(3))\cong\mathbb{Z}/2$, and $[\gamma]^2=[\gamma^2]=1$.