= Associative Lie product
{title2=$A^3=0$}
Over a <field> of <characteristic zero>, an associative Lie product is a <bilinear map> that is simultaneously an <associative algebra> product and a <Lie bracket>. Such a product has every triple product zero. To prove this, write multiplication by juxtaposition. By <associativity> and <antisymmetry of a Lie bracket>,
$$
xyz=-xzy=zxy,
$$
since moving $z$ left through $y$ and then through $x$ introduces two minus signs. Applying <antisymmetry of a Lie bracket> instead to the two elements $xy$ and $z$ gives $(xy)z=-z(xy)$. Thus $xyz=-xyz$, and $A^3=0$ in <characteristic zero>.
Conversely, an <alternating multilinear map> defining a product whose triple products all vanish is associative, because both associations give zero, and satisfies the <Jacobi identity>, because each nested product gives zero. A nonzero example has <basis> $p,q,z$, products $pq=z=-qp$, and all other products of <basis vectors> zero. It is the <Heisenberg Lie algebra> with its <Lie bracket> used as the <associative algebra> product. Its image $A^2=kz$ annihilates $A$ on both sides, and it has no <identity element>.
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