Lie Isomorphisms in Prime Rings with Involution

Let R and R′ be prime rings with involutions of the first kind and with respective Lie subrings of skew elements K and K′. Furthermore assume ( RC : C) ≠ 1, 4, 9, 16, 25, 64, where C is the extended centroid of R. It is shown that any Lie isomorphism of K onto K′ can be extended uniquely to an associative isomorphism of 〈 K〉 onto 〈 K′〉, where 〈 K〉 and 〈 K′〉 are respectively the associative subrings generated by K and K′.