Invariance conditions on substructures of division rings

Cartan–Brauer–Hua Theorem is a well-known theorem which states that if [Formula: see text] is a subdivision ring of a division ring [Formula: see text] which is invariant under all elements of [Formula: see text] or [Formula: see text] for all [Formula: see text], then either [Formula: see text] or [Formula: see text] is contained in the center of [Formula: see text]. The invariance idea of this basic theorem is the main notion of this paper. We prove that if [Formula: see text] is a division ring with involution [Formula: see text] and [Formula: see text] is a subspace of [Formula: see text] which is invariant under all symmetric elements of [Formula: see text], then either [Formula: see text] is contained in the center of [Formula: see text] or is a Lie ideal of [Formula: see text]. Also, we show that if [Formula: see text] is a self-invariant subfield of a non-commutative division ring [Formula: see text] with a nontrivial automorphism, then [Formula: see text] contains at least one non-central proper subfield of [Formula: see text].