This paper describes the
structure of additive subgroups and of subrings which are invariant under Lie
commutation with higher commutators of the skew-symmetric elements in 2-torsion
free rings with involution. Except for cases arising when the subring is central, or
when the ring satisfies a polynomial identity of small degree, the invariant
subring must contain an ideal of the ring. With the same exceptions, the
invariant subgroup must contain either the derived Lie ring of the set of
skew-symmetric elements in some ideal, or the Lie product of the set of
skew-symmetric elements in the ideal with the set of symmetric elements in the
ideal. Furthermore, the appropriate one of these Lie products is not Lie
solvable.
