In this article, we investigate factorization problems for twisted triple product Galois representations over real quadratic fields, arising from families of Hilbert cusp forms. Specifically, we address the factorization in two distinct settings determined by the order of vanishing of associated L-unctions at their central critical values-namely, the rank (1, 1) and rank (0, 2) cases. Our results generalize the algebraic factorization framework developed by Büyükboduk et al. in higher rank scenario to the setting of real quadratic fields. Notably, our work yields the first known factorization result in the higher rank (0, 2) case, marking a significant advancement in the study of triple product motives over totally real fields.
Das et al. (Mon,) studied this question.