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tokens: we present two other spectral extraction methods for the SPI data. B.1. Method 2: Using different event selection In this method, we consider the threshold of the PSD events, as set in the SPI telescope electronics. These are from 400 keV to 2.2 MeV after revolution 1170 and from nearly 650 keV to 2.2 MeV before. In this energy range, we used the PSD events only. This selection has been prescribed in Roques Jourdain (2019) in order to avoid spectral artifacts caused by potential spurious events. These spurious events appear at high energies (> 400 keV) for very bright sources (typically above 1 Crab). They are caused by a displacement of low-energy photons toward high energy, when the source is so bright that many photons can be detected in the same detection electronics time window. The energy reconstruction for the SE chain is then not reliable. This does not happen in the PE chain, which records the same event much more rapidly. B.2. Method 3: Using SPIDAI In this method, we use the SPI Data Analysis Interface 6 (SPIDAI) to reduce the data from SPI. We created a sky model containing Cyg X-1 and Cygnus X-3 and set the source variability to ten scws for Cygnus X-3 and five scws for Cyg X-1 (e.g., Bouchet et al. 2003 ; Cangemi et al. 2021 ). We then created the background model by setting the variability timescale of the normalization of the background pattern to five scws. In order to avoid the effects due to solar flares, radiation belt entries, and other nonthermal incidents, we removed scws for which the reconstructed counts compared to the detector counts give a χ 2 > 1.5. This selection reduces the total number of scws by ∼10%. The shadowgrams were then deconvolved to obtain the source flux, and spectra were extracted between 20 and 2000 keV using 27 logaritmically spaced channels. Given the large number of scws in both states, we extracted six and four different spectra for the LHS and HSS, respectively, each of them was extracted from data of 500–600 scws and corresponds to a distinct period. For each spectrum, we applied a correcting factor of 1/0.85 [above 400 keV or 650 keV according to the considered period, Roques Jourdain 2019 ) to account for the change efficiency. We then computed two average spectra for both states. In the HSS, the normalization of the spectrum has a slight variability over the four different periods which explain the larger error bars that we observe in this state when computing the average spectrum. B.3. Comparison between the methods The right panel on Fig. B.1 shows the broad SPI spectra (20–2000 keV) used in this paper along with the spectra extracted with the aforementioned methods. The left panel on Fig. B.1 shows the same spectra between 400 keV and 700 keV, that is to say the range for which the selection between SEs and PSD events is different from the the method described in this paper and method 2, described in Appendix B.1 . Fig. B.1. Left panel: spectra extracted with three different methods: