matrices - Element of $mathfrak{sl}(2,Bbb C)$ corresponding to element in $SL(2,Bbb C)$

If I have $\begin{bmatrix}a&0\\0&a^{-1}\end{bmatrix}$ in $SL(2,\Bbb C)$, how do I find what element I would have corresponding to this in $\mathfrak{sl}(2,\Bbb C)$? I imagine it might be something like $\begin{bmatrix}a&0\\0&-a\end{bmatrix}$, but I am not sure how to find this.

I know I want to go from determinant $1$ matrices to traceless matrices. But I can't get the correspondence down yet.

Answer

The exponential map $\exp: \mathfrak{g}\rightarrow G$ gives you the matrices, however it need not be surjective (or injective) in general. Indeed, for $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$ and $G=SL_2(\mathbb{C})$ it is not - see here, or here. However, the diagonal matrices have preimages in the Lie algebra, as was shown already. So you can find such matrices of trace zero.