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 (Duke Math. J. $\bf{26}$ (1959), 623 – 628, DOI: 10.1215/S0012-7094-59-02660-2) 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?