Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part II

Subsequent to the previous paper [Tak5], we are concerned with the classification of complex prime $\mathbb{Q}$-Fano $3$-folds of anti-canonical codimension 4 which are produced, as weighted complete intersections of appropriate weighted projectivizations of certain affine varieties related with ${\mathbb{P}}^2\times {\mathbb{P}}^2$-fibrations. Such affine varieties or their appropriate weighted projectivizations (possibly allowing some coordinates have weights $0$) are called key varieties for prime $\mathbb{Q}$-Fano 3-folds. The purpose of this paper is to give new constructions of a $14$-dimensional affine variety ${\mathrm{{\rm Y}}}_{\mathbb{A}}^{14}$ and a $15$-dimensional affine variety ${\mathrm{\Pi }}_{\mathbb{A}}^{15}$ related with ${\mathbb{P}}^2\times {\mathbb{P}}^2$-fibration, which were constructed and were shown to be key varieties in the papers [Tak3,Tak4] and [Tay]. It is well-known that the affine cone of the Segre embedded ${\mathbb{P}}^2\times {\mathbb{P}}^2$ is defined as the null loci of the so called $\mathrm{♯}$-mapping of a 9-dimensional nondegenerate quadratic Jordan algebra $J$ of a cubic form. Inspired with this fact, we construct ${\mathrm{{\rm Y}}}_{\mathbb{A}}^{14}$ and ${\mathrm{\Pi }}_{\mathbb{A}}^{15}$ in the same way coordinatizing $J$ with 9 and 10 parameters, respectively. The coordinatization with $9$ parameters is derived by using a fixed primitive idempotent, and the associated Peirce decomposition. The coordinatization with $10$ parameters is derived from the construction of a quadratic Jordan subalgebra generated by $\mathrm{♯}$-products of two elements due to Petersson [Pe1].