  A Banach algebra $A$ is said to be zero Lie product determined if every
continuous bilinear functional $\varphi \colon A\times A\to \mathbb{C}$
satisfying $\varphi(a,b)=0$ whenever $ab=ba$ is of the form
$\varphi(a,b)=\omega(ab-ba)$ for some $\omega\in A^*$. We prove that $A$ has
this property provided that any of the following three conditions holds: (i)
$A$ is a weakly amenable Banach algebra with property $\mathbb{B}$ and having a
bounded approximate identity, (ii) every continuous cyclic Jordan derivation
from $A$ into $A^*$ is an inner derivation, (iii) $A$ is the algebra of all
$n\times n$ matrices, where $n\ge 2$, over a cyclically amenable Banach algebra
with a bounded approximate identity.
