I am reading the book "Introduction to mechanics and symmetry" by J.Marsden and T.Ratiu and am experenced a problem.

Let $(P,\Omega)$ be a symplectic manifold, a submanifold $S\subset P$ is called a symplectic submanifold when $\omega:=i^*\Omega$ is a symplectic form on $S, i:S\rightarrow P$ being the inclusion. Assume that $\dim P=2n,\dim S=2k$. In a neighborhood of a point $z_0\in S$, choose coordinates $z^1,...,z^{2n}$ on $P$ such that $S$ is given by $$z^{2k+1}=0,...,z^{2n}=0,$$ and so $z^1,...,z^{2k}$ provide the local coordinates for $S$.

In the formulation of Dirac's bracket formula, it appears the inverse of the matrix defined by $$C^{ij}(z)=\{z^i,z^j\},\ i,j=2k+1,...,2n.$$

The author said that it is easy to see one can choose coordinates such that the matrix $C=\{C^{ij}\}$ is invertible. But I cannot see the reason.

Thank you if you can give a short proof or refer me to a good reference.

Chengbo