The family in $\mathbb{P}^4$ given by the non-complete intersection
$x_0^2 + a x_2 x_3 - a^{-1} x_1 x_4 = 0$
$x_1^2 + a x_3 x_4 - a^{-1} x_2 x_0 = 0$
$x_2^2 + a x_4 x_0 - a^{-1} x_3 x_1 = 0$
$x_3^2 + a x_0 x_1 - a^{-1} x_4 x_2 = 0$
$x_4^2 + a x_1 x_2 - a^{-1} x_0 x_3 = 0$
The family in$\mathbb{P}^4$ given by the non-complete intersection
$x_0^2 + a x_2 x_3 - a^{-1} x_1 x_4 = 0$
$x_1^2 + a x_3 x_4 - a^{-1} x_2 x_0 = 0$
$x_2^2 + a x_4 x_0 - a^{-1} x_3 x_1 = 0$
$x_3^2 + a x_0 x_1 - a^{-1} x_4 x_2 = 0$
$x_4^2 + a x_1 x_2 - a^{-1} x_0 x_3 = 0$