Does the following system of six simultaneous equations in eight variables $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4$ have solutions in $\mathbb{R}$? in $\mathbb{C}$? $x_1y_2-x_2y_1=1$ $x_1y_3-x_3y_1=0$ $x_1y_4-x_4y_1=0$ $x_2y_3-x_3y_2=0$ $x_2y_4-x_4y_2=0$ $x_3y_4-x_4y_3=1$
Maple cannot help; I don't know if Mathematica will.