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

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 $\mathbb{RP}^{3}$. Here some generators of the subgroup of $Cr_3 (\mathbb{R})$ are determined, whose birational maps are automorphisms of the quartic surface.