Seismic observations have suggested the presence of two Large Low Shear Velocity Provinces (LLSVPs) in the lowermost mantle whose nature and origin are still debated. Several studies have tried to infer their potential composition using seismic observations with the hope to identify their formation mechanism. In particular, compositions enriched in iron (similar to 12-14 wt%) and bridgmanite (similar to 90 vol%) have been identified as promising candidates. Interestingly, these characteristics are somewhat consistent with the cumulates produced by the solidification of a primitive magma ocean, except that the iron enrichment should be much larger (> 20 wt%). Here, we provide a reappraisal of potential LLSVPs compositions based on an improved mineralogical model including, in particular, the effects of alumina. We systematically investigate the effects of six parameters: FeO and Al2O3 content, proportion of CaSiO3 and bridgmanite (so that the proportion of ferropericlase is implicitly investigated), Fe3+/Sigma 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 Al2O3 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 (<6 vol%); (ii) the formation of Fe3+-bearing bridgmanite is much more favored compared to other iron-bearing components. Following these results, we identify two endmember compositions: a Bm-rich and a CaPv-rich one. For each end-member composition, a large range of parameters is possible. We note, however, that a low temperature contrast (<500 K) is favored, and that a certain proportion between FeO content, Al2O3 and oxidation state should be maintained. 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 Al2O3 contents 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. (C) 2020 Elsevier B.V. All rights reserved.