  In this article, we establish an $\ell^2$ decoupling inequality for the
surface $$F_4^2:=\Big\{(\xi_1,\xi_2,\xi_1^4+\xi_2^4): (\xi_1,\xi_2) \in
[0,1]^2\Big\}$$ associated with the decomposition adapted to finite type
geometry from our previous work. The key ingredients of the proof include the
so-called generalized rescaling technique, an $\ell^2$ decoupling inequality
for the surfaces $$\Big\{(\xi_1,\xi_2,\phi_1(\xi_1)+\xi_2^4): (\xi_1,\xi_2) \in
[0,1]^2\Big\}$$ with $\phi_1$ being non-degenerate, reduction of dimension
arguments and induction on scales.
