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An extended derivation (endomorphism) of a (restricted) Lie algebra $L$ is anassignment of a derivation (respectively) of $L'$ for any (restricted) Liemorphism $f:L\to L'$, functorial in $f$ in the obvious sense. We show that (a)the only extended endomorphisms of a restricted Lie algebra are the two obviousones, assigning either the identity or the zero map of $L'$ to every $f$; and(b) if $L$ is a Lie algebra in characteristic zero or a restricted Lie algebrain positive characteristic, then $L$ is in canonical bijection with its spaceof extended derivations (so the latter are all, in a sense, inner). Theseresults answer a number of questions of G. Bergman.  In a similar vein, we show that the individual components of an extendedendomorphism of a compact connected group are either all trivial or all innerautomorphisms.

Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras