Here, we provide a reappraisal of potential LLSVPs compositions based on an improved mineralogical model including, for instance, the effects of alumina. We also systematically investigate the effects of six parameters: FeO and Al$_{2}$O$_{3}$ content, proportion of CaSiO$_{3}$ and bridgmanite (so that the proportion of ferropericlase is implicitly investigated), Fe$^{3+}$/$sum$Fe and temperature contrast between far-field mantle and LLSVPs. From the 81 millions cases studied, only 79000 cases explain the seismic observations. Nevertheless, these successful cases involve a large range of parameters with, for instance, FeO content between 12--25~wt% and Al$_{2}$O$_{3}$ content between 3--17~wt%. We then apply a principal component analysis (PCA) to these cases and find two robust results: (i) the proportion of ferropericlase should be low ($<$6vol%); (ii) the formation of Fe$^{3+}$-bearing bridgmanite is much more favored than other iron-bearing phases. Following these results, we identify two end-member compositions, Bm-rich and CaPv-rich, and discuss their characteristics. Finally, we discuss different scenarios for the formation of LLSVPs and propose that investigating the mineral proportion produced by each scenario is the best way to evaluate their relevance. For instance, the solidification of a primitive magma ocean may produce FeO and Al$_{2}$O$_{3}$ content similar to those suggested by our analysis. However, the mineral proportion of such reservoirs is not well-constrained and may contain a larger proportion of ferropericlase than what is allowed by our results.
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