Trace identities are invariant under simultaneous conjugation.
They are frequently used in the invariant theory of
matrices to find the generators and relations of the ring of invariants, and therefore are useful in answering questions similar to that posed by Hilbert's fourteenth problem.
Rowen, Louis Halle (2008), Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, vol.
2, American Mathematical Society, p. 412, ISBN 9780821841532.