TITLE: Is there any example of a Lie algebra which is not a derivation algebra?
QUESTION [6 upvotes]: I'm just studying Lie algebras. If $A$ is a $k$-algebra (not necessarily Lie or associative, just a bilinear law), it is straightforward to check that any derivation algebra of $A$ is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample.

Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any $k$-algebra (again, not necessarily associative or unital)?

Is there any example of a Lie algebra which is not isomorphic to the derivation algebra of any Lie algebra?


For the the second question the user YCor gives a positive answer. However, I am more interested in the first (and actually my original) question. Thank you!

REPLY [6 votes]: If $A$ is an arbitrary algebra over a field of characteristic $p>0$, then its derivation algebra is a restricted Lie algebra under the usual Lie bracket $[D_1,D_2]=D_1D_2-D_2D_1$ and the ordinary $p$-exponentation. This shows that, in positive characteristic, a non-restrictable Lie algebra cannot be the derivation algebra of an algebra of any kind.