= Antisymmetry of a Lie bracket
{title2=$[x,y]=-[y,x]$}
The relation $[x,y]=-[y,x]$ for a <bilinear map> proposed as a <Lie bracket>. An <Alternating Lie bracket> always satisfies it. Conversely, setting $y=x$ gives $2[x,x]=0$, so antisymmetry implies alternation over a <field> of <characteristic> different from two.
In <characteristic two>, the sign disappears and this relation just says $[x,y]=[y,x]$. It does not force $[x,x]=0$, even when the <Jacobi identity> holds. For example, on a two-dimensional <vector space> with <basis> $u,v$, set $[u,u]=v$ and every other <basis> bracket to zero, then extend bilinearly. Every nested bracket vanishes, so the <Jacobi identity> holds, and the product is symmetric. Nevertheless it is not a <Lie algebra>, because $[u,u]\ne0$.
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