We revisit the Kahan–Hirota–Kimura discretization of a quadratic vector field. The corresponding discrete system is generated by successive iterations of a birational map Fh. We include a proof of a formula for the Jacobian of this map. In the following, we essentially focus on the case of the Lotka–Volterra system. We discuss the notion of focal points and prefocal lines of the map Fh and of its inverse Fh−1. We show that the map Fh is the product of two involutions. The nature of the fixed points of Fh is studied. We introduce the notion of asymptotic focal and prefocal sets. We further provide a new proof of the theorem of Sanz-Serna. We show that the mapping Fh is integrable for h=1 and that it preserves a pencil of conics (generic hyperbolas). To conclude, we provide several numerical simulations for 0 < h < 1.