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

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.