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".

**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?