# Which known theorems of Lie algebras are still valid for Leibniz algebras?

Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras to Leibniz algebras. For instance, (1) gives a generalization of Engel's theorem to Leibniz algebras.

Is there a survey of which known theorems of Lie algebras are still valid (and also not valid) for general Leibniz algebras? If not, could we make a community wiki to gather examples? I think it would be a nice idea to put these examples in the article of Wikipedia.

(1): Ayupov, Sh A., and B. A. Omirov. "On Leibniz algebras." Algebra and operator theory. Springer, 1998. 1-12.

• One example: if a Leibniz algebra (over any commutative ring) admit a grading in $\mathbf{Z}$ such that $\mathfrak{g}_n=0$ for all $|n|\ge n_0$ for some $n_0$ and $\mathfrak{g}_0=0$, then $\mathfrak{g}$ is nilpotent. (See my answer to mathoverflow.net/questions/90964/… )
– YCor
Dec 29, 2014 at 16:36
• Note that this nilpotency result is trivial for positive gradings ($\mathfrak{g}_n=0$ for all $n<0$ and all $n\ge n_0$), but the latter trivially holds for all algebras (module + bilinear law) while the former doesn't, with simple counterexamples.
– YCor
Dec 29, 2014 at 16:43