Given the Adjoint action of a matrix;
$\text{Ad}(g) X_1 = g \, X_1 \, g^{-1} = X_2 $.
Where g is in a (matrix) Lie group, $X_1,\; X_2$ are from the Lie algebra, can a $g$ be written in terms of the given $X_1$ and $X_2$?
Specifically - apart from solving the system of equations you obtain, or doing numerical calculations - are there any known results?
First, assume that $X_2=X_1$ then we have that $Ad(g).X_1=X_1 \Leftrightarrow g\in Z_G(\exp(X_1))$, hence in that case there are as many solution as there are elements in some centralizer. Furthermore, I claim that if you have any $X_1$ and $X_2$ such that $Ad(g_0).X_1=X_2$ then the set of $g$ such that $Ad(g).X_1=X_2$ is given by $g_0Z_G(\exp(X_1))$.
Hence to answer your question, you need to look for an expression for the centralizer of $\exp(X_1)$ which is given quite explicitely once you know the Jordan form of the matrix $\exp(X_1)$.