Let $L$ be a Lie algebra and denote by $\mathrm{Der}(L)$ the derivation algebra of $L$. If $L$ is finite-dimensional, then a theorem of Leger and Togo states that  $\mathrm{Der}(L)$ is nilpotent if and only if $L$ is characteristically nilpotent or 1-dimensional. (I recall that a Lie algebra is said to be characteristically nilpotent if all of its derivations are nilpotent linear transformations). 

 Is there any result of this sort in the infinite-dimensional case?
What can be said on the structure of $L$ when $\mathrm{Der}(L)$ is nilpotent (besides the obvious fact that $L$ is nilpotent)?