Field's characteristic in Lie algebras

Question

If $L$ is a Lie algebra over a field $\mathbb { F}$ with characteristic not 2 , $[x, x]=0 , \forall x\in L$ if and only if $[ x, y]=- [ y, x] ,\forall x, y\in L$. I would like an example of a (non-associative) algebra (necessarily in characteristic 2) in which the Jacobi identity and the antisymmetric property both hold, but that is not a Lie algebra (that is, the product is not alternating).

Answer 1

Char 2. Basis $(e_1,e_2)$. Bracket: $[e_1,e_1]=e_2$, all other basis brackets being zero.

It's symmetric, so antisymmetric since characteristic is 2. All triple brackets are zero (2-nilpotent algebra) and thus Jacobi identity holds. But by definition the bracket is not alternating.