Construct a outer derivation for every nonzero nilpotent/solvable Lie algebra

Question

A derivation is inner if it has the form $\text{ad}x$, $x\in L$. Otherwise, a derivation is outer. For a semisimple Lie algebra, every derivation is inner. What about the nilpotent and solvable cases?

Answer 1

Every nonzero nilpotent Lie algebra over an arbitrary field has an outer derivation. This was proved by Dixmier, and by Jacobson and Schenkmann in $1955$.

This is no longer true for solvable Lie algebras. Already the nonabelian $2$-dimensional Lie algebra $\mathfrak{r}_2(K)$ has only inner derivations.

Reference: Why, for nilpotent Lie Algebras, is the inclusion to the derivations $x \mapsto {ad}_x$ not surjective?

Yves has written down there the proof for nilpotent Lie algebras.

Note: It is not true, that all derivations of semisimple Lie algebras are inner. Consider the simple Lie algebra $L=\mathfrak{psl}_3(\Bbb F_3)$. Then we even have $\operatorname{Out}(L)\cong L$, see here.