The category of Lie algebras is equivalent to a certain category of cocommutative Hopf algebras, with the equivalence given by sending a Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\mathfrak{g})$. These cocommutative Hopf algebras can in turn be thought of as group objects in a certain category of cocommutative coalgebras, and hence can potentially pop up as automorphism objects in any category enriched over cocommutative coalgebras. 

You might object that you don't know any interesting examples of such categories, but in fact you do: the category of *commutative algebras* admits such an enrichment (see <a href="http://ncatlab.org/nlab/show/measuring+coalgebra">the nLab</a>), and this is one abstract way to see why Lie algebras can act on commutative algebras (by derivations).