I recently come across the equation of the form:
$$ 0 = - 5 + 4(x^2+y^2+z^2+w^2) + 16 x y z w$$
As can be seen, it is quartic in four variables $x,y,z,w$. This yields a three-dimensional surface in a four-dimensional space. To visualize four-dimensional space, just like in four-dimensional spacetime, we can consider $w$ a "time variable" and study the three-dimensional surfaces $(x,y,z)$ for fixed $w$.
However, my question is, whether the above surface has a name and whether it has been studied somewhere (i.e. solution of the above polynomial equations)? For e.g., when $w =1,-1$, I suspect that it is related to the "Cayley nodal cubic surface".