Commuting traces of biadditive maps revisited

Let R be a prime ring with char (R) not equal 2. It is known that the form of a biadditive map B: R x R --> R satisfying [B(x, x), x] = 0 for all x is an element of R can be described provided that R does not satisfy the standard polynomial identity S-4. In this note we prove a linear algebraic result from which we are able to derive that the description of B is the same even when R satisfies S-4. This makes it possible to complete the results on the structure of Lie isomorphisms and Lie derivations on prime rings.