On the addition of an $SU(2)$ quadruplet of scalars to the Standard Model

We consider the extension of the Standard Model through an $SU(2)$ quadruplet of scalars with hypercharge either $0$, $3/2$, or $1/2$ (with an additional reflection symmetry in the latter case). We establish, through exact analytical equations, the boundaries of the phase spaces of the gauge-invariant terms that appear in the (renormalizable) scalar potentials. We devise procedures for the determination of necessary and sufficient bounded-from-below conditions on those potentials.