The Lie operad $\text{Lie}$ is generated by a binary operator $[\ ,\ ]$, modulo a degree two relation (skew commutativity $[x,y]=-[y,x]$) and a degree three relation (Jacobi $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]$). See e.g. the paper ["The operad Lie is free"](https://arxiv.org/pdf/0802.3010.pdf).


**Question:** do "non(skew)commutative Lie algebras" ever show up in nature? i.e. is there an operad $\text{Lie}^{\mathrm{nc}}$ generated by a binary operator $[\ ,\ ]$ subject to Jacobi *and some subset of the relations generated by Jacobi and skew-commutativity* (so there is a map $\text{Lie}^{\mathrm{nc}}\to \text{Lie}$) *but not including skew-commutativity*, and so that its algebras $\text{Alg}\_{\text{Lie}^{\mathrm{nc}}}$ show up in e.g. algebra or geometry?



Non(skew)commutative Lie algebras?
Such objects are known as *Leibniz algebras*.
A Leibniz algebra is a module $M$ together with a bilinear pairing $$[-,-]\colon M⊗M→M$$ that satisfies the Leibniz identity: $$[a,[b,c]]=[[a,b],c]+[b,[a,c]].$$


Leibniz algebras for which $[a,b]=-[b,a]$ are precisely Lie algebras.


The concept was introduced and studied by A. Blokh: [A generalization of the concept of a Lie algebra](https://www.mathnet.ru/eng/dan/v165/i3/p471).


Later, it was studied by Loday, who also coined the name “Lebniz algebra”: [Une version non commutative des algèbres de Lie: les algèbres de Leibniz](http://www.numdam.org/item/RCP25_1993__44__127_0/).


The article [Leibniz algebra](https://ncatlab.org/nlab/show/Leibniz+algebra) contains many more additional references.


