Isolations of the sum of two squares from its proper subforms

For a (positive definite and integral) quadratic form f, a quadratic form is said to be an isolation of f from its proper subforms if it represents all proper subforms of f, but not f itself. It was proved that the minimal rank of isolations of the square quadratic form $x^2$ is three, and there are exactly 15 ternary diagonal isolations of $x^2$. Recently, it was proved that any quaternary quadratic form cannot be an isolation of the sum of two squares $I_2=x^2+y^2$, and there are quinary isolations of $I_2$. In this article, we prove that there are at most 231 quinary isolations of $I_2$, which are listed in Table 1. Moreover, we prove that 14 quinary quadratic forms with dagger mark in Table 1 are isolations of $I_2$.