Jordan maps and zero Lie product determined algebras

Let $A$ be an algebra over a field $F$ with {\rm char}$(F)\mathrm{\ne }2$. If $A$ is generated as an algebra by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\mathrm{\Phi }:A\times A\to X$, where $X$ is an arbitrary vector space over $F$, the condition that $\mathrm{\Phi }(x^2,x)=0$ for all $x\in A$ implies that $\mathrm{\Phi }(xy,z)+\mathrm{\Phi }(zx,y)+\mathrm{\Phi }(yz,x)=0$ 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.