The function $\exp: \mathfrak{su}(2)\to SU(2)$ is surjective.
I know that $SU(2)\cong\mathbb{S^3}$ and that $\mathbb{S^3}=\{x\in \mathbb{H}:|x|=1\}$ where $\mathbb{H}=\left \langle 1,i,j,k \right \rangle$, I also know that every element $A\in\mathfrak{su}(2) $ can be written as $A=\begin{bmatrix} bi& c+di\\ -c+di & -bi\end{bmatrix}$ where $b,c,d\in\mathbb{R}$, but I don't know what the preimage of an arbitrary $B$ element in $SU(2)$ would be, could someone please help me with any ideas, thank you very much.
Any unitary matrix is normal, so it can be diagonalized by a unitary matrix. Thus if $A \in SU(2)$, we can write $$ A = S \pmatrix{e^{i\theta} & 0\cr 0 & e^{-i\theta}} S^*$$ with $S \in U(2)$, and this is $\exp(B)$ where $$ B = S \pmatrix{i\theta & 0\cr 0 &-i\theta} S^* \in {\frak su}(2)$$