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In classical invariant theory, the Gr\"obner base of the ideal of syzygiesand the normal forms of polynomials of invariants are two core contents. Toimprove the performance of invariant theory in symbolic computing of classicalgeometry, advanced invariants are introduced via Clifford product. This paperaddresses and solves the two key problems in advanced invariant theory: theGr\"obner base of the ideal of syzygies among advanced invariants, and thenormal forms of polynomials of advanced invariants. These results beautifullyextend the straightening of Young tableaux to advanced invariants.

Normalization of Polynomials in Algebraic Invariants of Three-Dimensional Orthogonal Geometry