Somos-4 and a quartic Surface in ${\mathbb{R}\mathbb{P}}^3$

The Somos-4 equation defines the sequences with this name. Looking at these sequences with an additional property we get a quartic polynomial in 4 variables. This polynomial defines a rational, projective surface in ${\mathbb{R}\mathbb{P}}^3$. Here some generators of the subgroup of $C{r}_3(\mathbb{R})$ are determined, whose birational maps are automorphisms of the quartic surface.