Consider $T,S: \mathfrak{g} \longrightarrow \mathfrak{g}$ automorphisms of the $n$-dimensional Lie algebra $\mathfrak{g}$. Is there any relation between the eigenvalues of $T$ and $S$ and the eigenvalues of the automorphism $T \circ S: \mathfrak{g} \longrightarrow \mathfrak{g}$ (or vice-versa)?
The specific problem I'm dealing with is considering $T = e^{\hbox{ad}(X)}$ and $S = e^{\hbox{ad}(Y)}$.
I believe that the relation may not exist, but I'm not sure.
Any hint, reference or counterexample is appreciated.