Inner automorphisms of a Lie algebra are typically defined as automorphisms generated by elements of the form $exp (ad_X)$ where $X$ is nilpotent. Is $exp (ad_X)$ not inner for $X$ having a non-trivial Jordan-Chevalley decomposition?
This question is important to classification questions, which are often up to inner automorphism; classifying subalgebras up to inner automorphism, for instance. Certainly one needs to understand the definition of inner automorphism if one hopes to undertake such a classification.