General Bivector Projection of the Rotor Loop

Determine the bivector projection of the same-order rotor loop Θ_k = (U_1 ⋯ U_k)(\widetilde U_1 ⋯ \widetilde U_k) for general k, in order to obtain the order defect as a rotation plane rather than only through its scalar part.

Background

The paper derives a closed formula for the scalar projection of the same-order rotor loop Θ_k. It notes that higher-grade projections are also determined by the family, and gives the complete decomposition for k = 2. The first unresolved higher-grade problem concerns the bivector component for arbitrary k; the authors identify this as potentially useful because it would provide geometric rotation-plane information in addition to the scalar order defect. Numerical observations for small k motivate the question but do not resolve it.