  In classical invariant theory, the Gr\"obner base of the ideal of syzygies
and the normal forms of polynomials of invariants are two core contents. To
improve the performance of invariant theory in symbolic computing of classical
geometry, advanced invariants are introduced via Clifford product. This paper
addresses and solves the two key problems in advanced invariant theory: the
Gr\"obner base of the ideal of syzygies among advanced invariants, and the
normal forms of polynomials of advanced invariants. These results beautifully
extend the straightening of Young tableaux to advanced invariants.
