If $K(s,t)$ is a local solution to the split Monge-Ampère equation (17), then $\Psi(y)$ is a (local) solution of the classical (real) Monge-Ampère equation

(19)

$$\det\frac{\partial^{2}\Psi}{\partial y_{i}\partial y_{p}}=1,\quad 1\leq i,j\leq n,$$

where

(20)

$$y_{i}=\frac{\partial K}{\partial s_{i}},\ 1\leq i\leq k,\qquad y_{p}=t_{p},\ k+1\leq p\leq n,$$

and $\Psi(y)$ is the partial Legendre transform of $K(s,t)$, defined by

(21)

$$\frac{\partial\Psi}{\partial y_{i}}=s_{i},\ 1\leq i\leq k,\qquad\frac{\partial\Psi}{\partial y_{p}}=-\frac{\partial K}{\partial t_{p}},\ k+1\leq p\leq n.$$