We prove that, over arbitrary base schemes, quintic del Pezzo threefolds are classified by non-degenerate ternary symmetric bilinear forms. As applications, we describe the automorphism groups, the Hilbert schemes of lines, the orbit decompositions, and the normalizations of the orbit closures of quintic del Pezzo threefolds, by placing special emphasis on the case of characteristic two, where symmetric bilinear forms behave differently. Furthermore, we refine a known result on arithmetic finiteness of quintic del Pezzo threefolds.
Ito et al. (2025) studied this question.