The exponential is not injective: rotations as exponentials

Powers of $J$. $J^2=-I$, hence $J^3=-J$, $J^4=I$, and in general $J^{2k}=(-1)^kI$, $J^{2k+1}=(-1)^kJ$.

Splitting the series. Separating even and odd terms,
$$e^{\theta J}=\sum_{k\ge0}\frac{\theta^{2k}}{(2k)!}J^{2k}+\sum_{k\ge0}\frac{\theta^{2k+1}}{(2k+1)!}J^{2k+1}=\Bigl(\sum_k\frac{(-1)^k\theta^{2k}}{(2k)!}\Bigr)I+\Bigl(\sum_k\frac{(-1)^k\theta^{2k+1}}{(2k+1)!}\Bigr)J.$$
The two scalar series are $\cos\theta$ and $\sin\theta$, so
$$e^{\theta J}=\cos\theta\,I+\sin\theta\,J=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}=R_\theta.$$
(Rearranging the series into even and odd parts is allowed by absolute convergence.)

Non-injectivity. $e^{2\pi J}=R_{2\pi}=I=e^{0}$ but $2\pi J\ne0$. More generally $e^{(\theta+2\pi k)J}=e^{\theta J}$ for all integers $k$.

Remark. This is the matrix version of $e^{i\theta}=\cos\theta+i\sin\theta$: the subalgebra $\{aI+bJ\}$ of $M_2(\mathbb{R})$ is isomorphic to $\mathbb{C}$, with $J\leftrightarrow i$. Non-injectivity of $\exp$ on the real matrices is the same phenomenon as the multivaluedness of the complex logarithm.