Lie algebras over a field

Let $k$ be a field. A Lie algebra over $k$ is a $k$-vector space $\mathfrak{g}$ (Vector space over a field) together with a map

$[-,-]:\mathfrak{g}\times \mathfrak{g}⟶\mathfrak{g}$

that is linear in each variable (Linear map between vector spaces over the same field) and satisfies, for all $x,y,z\in \mathfrak{g}$,

$[x,x]=0$

and

$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.$

The first identity is alternation and the second is the Jacobi identity. No finite-dimensional hypothesis is imposed. Alternation implies $[x,y]=-[y,x]$ in every characteristic: expand $0=[x+y,x+y]=[x,y]+[y,x]$. Thus skew-symmetry is a consequence here, not a replacement for alternation in characteristic $2$.

The zero vector space, with its unique bracket, is a Lie algebra. A Lie algebra is abelian when $[x,y]=0$ for every $x,y$.