On the bounded-from-below conditions for the extension of the SM through a scalar quadruplet

We study the scalar potential of the extension of the Standard Model through a scalar quadruplet with hypercharge either 1/2 or 3/2. That potential contains a term that had not been previously taken into account. We write down the full unitarity conditions. We use a numerical scan of phase space instead of attempting to derive fully analytical bounded-from-below conditions.