what will be the exponential map from the Lie algebra $sl(2, C) \ltimes_{ad} sl(2,C)$ to its Lie group $Sl(2, C) \ltimes_{Ad} Sl(2,C)$.

Question

Let $G:=Sl(2, C) \ltimes_{Ad} Sl(2,C)$, where $Ad_X(Y)=XYX^{-1}$ for all $X, Y \in SL(2,C)$. It is known that $G$ is a Lie group and its Lie algebra is given by $g:=sl(2, C) \ltimes_{ad} sl(2,C)$, where $ad_x(y)=[x,y]=xy-yx$ for all $x, y \in sl(2,C)$. Its exponential map is given by $Exp: g \rightarrow G$, by $Exp(x,y)=(e^x, \phi(x,y))$, where $\phi$ is map from $g$ to $Sl(2,C)$. Is it possible to calculate the map $\phi$ explicitly in this case?