Alternating Lie bracket ($[x,x]=0$)

A Lie bracket is alternating when $[x,x]=0$ for every $x$. This condition is imposed in every characteristic of a field. Bilinearity gives $0=[x+y,x+y]=[x,y]+[y,x]$, hence antisymmetry of a Lie bracket. Requiring alternation directly keeps the definition correct in characteristic two.