I am reading the paper A nilpotent group and its elliptic curve: Non-uniformity of local zeta functions of groups by du Sautoy. In this paper he defines the nilpotent group $$ G=\left\langle x_1,x_2,x_3,x_4,x_5,x_6,y_1,y_2,y_3\Big\rvert \begin{matrix} [x_1,x_4]=y_3,[x_1,x_5]=y_1,[x_1,x_6]=y_2, [x_2,x_4]\\=y_1, [x_2,x_5]=y_3, [x_3,x_4]=y_2,[x_3,x_6]=y_1 \end{matrix} \right\rangle $$ where all other commutators are 1. He proceeds to claim that this group is related to the elliptic curve $Y^2=X^3-X$. This should become clear by computing the following determinant: $$ \begin{vmatrix} [x_1,x_4] & [x_2,x_4] & [x_3,x_4] \\ [x_1,x_5] & [x_2,x_5] & [x_3,x_5] \\ [x_1,x_6] & [x_2,x_6] & [x_3,x_6] \\ \end{vmatrix} = \begin{vmatrix} y_3 & y_1 & y_2\\ y_1 & y_3 & 1 \\ y_2 & 1 & y_1 \\ \end{vmatrix}. $$ Unfortunately, I don't know how to compute the determinant when the values of the matrix are elements of some non-abelian group. I naively tried to compute it as if it was a 'normal' matrix, yielding $y_3^2y_1+y_1y_2+y_2y_1-y_2y_3y_2-y_3-y_1^3$. I am sure this is not the correct expression. Nevertheless, I don't see how an expression in three variables will give me insight in the relation of this group to $Y^2=X^3-X$ (two variables).
Does anyone know how to see this relation? Thanks in advance.