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Aims. We study the relative helicity of active region (AR) NOAA 12673 during a ten-hour time interval centered around a preceding X2.2 flare (SOL2017-09-06T08:57) and also including an eruptive X9.3 flare that occurred three hours later (SOL2017-09-06T11:53). In particular, we aim for a reliable estimate of the normalized self-helicity of the current-carrying magnetic field, the so-called helicity ratio, | H J |/| H 𝒱 |, a promising candidate to quantity the eruptive potential of solar ARs. Methods. Using Solar Dynamics Observatory Helioseismic and Magnetic Imager vector magnetic field data as an input, we employ nonlinear force-free (NLFF) coronal magnetic field models using an optimization approach. The corresponding relative helicity, and related quantities, are computed using a finite-volume method. From multiple time series of NLFF models based on different choices of free model parameters, we are able to assess the spread of | H J |/| H 𝒱 |, and to estimate its uncertainty. Results. In comparison to earlier works, which identified the non-solenoidal contribution to the total magnetic energy, E div / E , as selection criterion regarding the required solenoidal quality of magnetic field models for subsequent relative helicity analysis, we propose to use in addition the non-solenoidal contribution to the free magnetic energy, | E mix |/ E J , s . As a recipe for a reliable estimate of the relative magnetic helicity (and related quantities), we recommend to employ multiple NLFF models based on different combinations of free model parameters, to retain only those that exhibit smallest values of both E div / E and | E mix |/ E J , s at a certain time instant, to subsequently compute mean estimates, and to use the spread of the individually contributing values as an indication for the uncertainty. 

Context . In September 2017, the largest X-class flare of solar cycle 24 occurred from the most active region (AR) of this cycle, AR 12673. This AR attracted much interest because of its unique morphological and evolution characteristics. Among the parameters that were examined in the AR was magnetic helicity, but either only approximately, or intermittently, or both. Aims . We here study the evolution of the relative magnetic helicity and of the two components of its decomposition, the non-potential, and the volume-threading one, in the time interval around the highest activity of AR 12673. We especially focus on the ratio of the non-potential to total helicity, which has recently been proposed as an indicator of AR eruptivity. Methods . We first approximated the coronal magnetic field of the AR with two different optimization-based extrapolation procedures, and chose the method that produced the most reliable helicity value at each instant. Moreover, in one of these methods, we weighted the optimization by the uncertainty estimates derived from the Helioseismic and Magnetic Imager (HMI) instrument for the first time. We then followed an accurate method to compute all quantities of interest. Results . The first observational determination of the evolution of the non-potential to total helicity ratio seems to confirm the quality it has in indicating eruptivity. This ratio increased before the major flares of AR 12673 and afterwards relaxed to lower values. Additionally, we discuss the evolution patterns of the various helicity and energy budgets of AR 12673 and compare them with results from other works.