Let R be a prime ring with
involution and L a Lie ideal of R. It is known that if the commutator of an element of
R with L is in the center of R, then either the element or L is in the center, unless R
is an order in a simple algebra at most four dimensional over its center.
This result is shown to hold if L is replaced by its (skew-) symmetric part.
More generally, if a derivation of R sends the (skew-) symmetric part of L
into the center of R, then one of the three possibilities mentioned above
must hold. The same conclusion follows if one assumes that the image of L
under the derivation is contained in the set of (skew-) symmetric elements of
R.
