  Let $A$ be an algebra over a field $F$ with {\rm char}$(F)\ne 2$. If $A$ is
generated as an algebra by $[[A,A],[A,A]]$, then for every
  skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ is an arbitrary
vector space over $F$, the condition that $\Phi(x^2,x)=0 $ for all $x\in A$
implies that $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ for all $x,y,z\in A$. This
is applicable to the question of whether $A$ is zero Lie product determined,
and is also used in proving that a Jordan homomorphism from $A$ onto a
semiprime algebra $B$ is the sum of a homomorphism and an antihomomorphism.
