  We consider Bekenstein-Hawking entropy and attractors in extremal BPS black
holes of $\mathcal{N}=2$, $D=4$ ungauged supergravity obtained as reduction of
minimal, matter-coupled $D=5$ supergravity. They are generally expressed in
terms of solutions to an inhomogeneous system of coupled quadratic equations,
named BPS system, depending on the cubic prepotential as well as on the
electric-magnetic fluxes in the extremal black hole background. Focussing on
homogeneous non-symmetric scalar manifolds (whose classification is known in
terms of $L(q,P,\dot{P})$ models), under certain assumptions on the Clifford
matrices pertaining to the related cubic prepotential, we formulate and prove
an invertibility condition for the gradient map of the corresponding cubic form
(to have a birational inverse map which is an homogeneous polynomial of degree
four), and therefore for the solutions to the BPS system to be explicitly
determined, in turn providing novel, explicit expressions for the BPS black
hole entropy and the related attractors as solution of the BPS attractor
equations. After a general treatment, we present a number of explicit examples
with $\dot{P}=0$, such as $L(q,P)$, $1\leqslant q\leqslant 3$ and $P\geqslant
1$,or $L(q,1)$, $4\leqslant q\leqslant 9$, and one model with $\dot{P}=1$,
namely $L(4,1,1)$. We also briefly comment on Kleinian signatures and split
algebras. In particular, we provide, for the first time, the explicit form of
the BPS black hole entropy and of the related BPS attractors for the infinite
class of $L(1,P)$ $P\geqslant 2$ non-symmetric models of $\mathcal{N}=2$, $D=4$
supergravity.
