Jordan maps and zero Lie product determined algebras Dedicated to Vesselin Drensky on his 70th birthday

Let A be an algebra over a field F with char (F) not equal 2. If A is generated as an algebra by [[A, A], [A, A]], then for every skew-symmetric bilinear map Phi : AxA -> X, where X is an arbitrary vector space over F, the condition that Phi(x(2), x) = 0 for all x is an element of A implies that Phi(xy, z) + Phi(zx, y) + Phi(yz, x) = 0 for all x, y, z is an element of 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.