The origin of long gamma-ray bursts (GRBs) is thought to be related to the death of massive stars, being their association with type Ic supernovae (SNe) Galama et al. (1998); Woosley & Bloom (2006); Della Valle (2011); Hjorth & Bloom (2012) one of the most compiling observational evidence. Since most massive stars belong to binaries (see, e.g., Kobulnicky & Fryer 2007; Sana et al. 2012), the direct collapse of a massive star to a BH should not produce an SN, observed preSN progenitors have masses ≲ 18 M⊙ (Smartt 2009, 2015), and stellar evolution models predict the direct formation of a BH only in progenitor stars ≳ 25M⊙ (see, e.g., Heger et al. 2003), one can conclude that the GRB and the SN should not originate from a single star. Based on the above theoretical and observational clues, the binary-driven hypernova (BdHN) model proposes a binary system composed of a carbon-oxygen star (CO) and a neutron star (NS) companion as the progenitor of GRB-SNe. We refer the reader to Rueda & Ruffini (2012); Fryer et al. (2014, 2015); Becerra et al. (2016); Ruffini et al. (2018b,a); Becerra et al. (2019); Ruffini et al. (2019); Rueda & Ruffini (2020); Moradi et al. (2021); Ruffini et al. (2021); Rueda et al. (2022b,a); Wang et al. (2022); Rueda et al. (2022c); Becerra et al. (2022), for theoretical details and applications of the BdHN model to specific sources, and to Aimuratov et al. (2023) for the latest developments. The BdHN model assumes the occurrence of the SN onsets the entire cataclysmic event: the core collapse of the CO forms a newborn NS (hereafter νNS) and ejects material that accretes onto the NS companion and the νNS owing to matter fallback. In this picture, the binary’s orbital period is a critical parameter determining the system’s fate and energetics, which leads to the classification of BdHNe into type I (≳ 1052 erg), II (∼ 1050– 1052 erg), and III (≲ 1050 erg; see Aimuratov et al. 2023, for details). Since the SN trigger and the νNS formation are crucial in explaining the event, a vast new topic has emerged from studying alternatives to assuming a single nonrotating star core-collapse SN event in the CO core. The alternative scenario has been recently advanced based on the fission process of the CO core due to its high rotation rate gained by corotation with its NS companion. It has been indicated in Aimuratov et al. (2023) that the GRB-SN might be triggered, e.g., by a fast rotating 10M⊙ CO star set in corotation with a companion NS in an orbital period of a few minutes. The CO fission creates an 8.5M⊙ Maclaurin ellipsoid core and a 1.5M⊙ companion triaxial Jacobi ellipsoid (hereafter, JTE) in a Roche-lobe configuration (see Fig. 1). The aim of this article is to evaluate specific examples of the above fission process. For this task, we have generalized (see appendix A) the classical tables of the Maclaurin and Jacobi sequences in Chandrasekhar (1969); Jeans (1929) treatises necessary to describe the CO core fission in the present astrophysical scenario. In Section 2, we illustrate a few examples of the binary progenitor before fission, i.e., an initial 10M⊙ CO core corotating with an NS companion for selected values of the orbital period. Section 3 shows examples of the system after the fission of the initial 10M⊙ CO core into a Maclaurin new CO core of 8.5M⊙ and a JTE companion of 1.5M⊙ (see Fig. 1). Mass and angular momentum conservation are necessary conditions in the fission process. To exemplify, we assume the CO core after fission lies on a specific location of the Maclaurin sequence and the JTE, by definition, on the Jacobi sequence. One could select alternative locations for the fission products, and the outcomes would not vary significantly, as the physical quantities are similar. The five examples of fission we are considering are displayed in Tables 1 and 2. Table 1 lists the physical properties of the CO before fission, while Table 2 presents the physical properties of the fission products. The initial CO core in all examples has a mass of 10M⊙, while the rotation period varies.