It is very elementary; the graph representation refers to the cartesian product $P\times iP$ coming from the real vector space direct sum decomposition $\mathbb{C}^n = P\oplus iP\sim P\times iP$.

Here the direct sum decomposition is possible because $\dim _ \mathbb{R}(P)=n $ and $P\cap iP=0$, from the definition of Lagrangian subspace (thus any $z$ writes uniquely as $z=x+iy$ with $x$ and $y$ in $P$).

The second projection $\pi: P\times iP \to iP$, restricted to a subspace $Q$, has kernel $Q\cap iP$, so it is injective, hence bijective by dimension comparison, exactly for $Q\in U_P$, in which case $Q=\operatorname{graph}\\ (\pi_{|Q})^{-1}$.

For other details check e.g. the introductory pages of Hofer-Zehnder's book, Symplectic Invariants and Halmiltonian Dynamics.

It is very elementary; the graph representation refers to the cartesian product $P\times iP$ coming from the real vector space direct sum decomposition $\mathbb{C}^n = P\oplus iP\sim P\times iP$.

Here the direct sum decomposition is possible because $\dim _ \mathbb{R}(P)=n $ and $P\cap iP=0$, from the definition of Lagrangian subspace (thus any $z$ writes uniquely as $z=x+iy$ with $x$ and $y$ in $P$).

The second projection $\pi_2: P\times iP \to iP$, restricted to a subspace $Q$, has kernel $Q\cap iP$, so it is injective exactly for $Q\in U_P$, in which case $Q=\operatorname{graph}\\ \pi_1({\pi_2}{|Q})^{-1}$.

For other details check e.g. the introductory pages of Hofer-Zehnder's book, Symplectic Invariants and Halmiltonian Dynamics.