I was unsure if I should post this to the physics forum since mathematician often tend to think differently than physicists, but this is a claim about Lie algebras so I thought this forum might be more appropriate.
The real symplectic group is given by the set of matrices $2N \times 2N$ matrices s.t. ;
$$M^T \Omega M$$
with $\Omega = \begin{bmatrix}0 & 1_{N \times N}\\-1_{N \times N} & 0\end{bmatrix}$
Now I'm told that the rank of the Lie Algbebra (dimension of Cartan Subalgebra) $\mathfrak{sp}(2N,R)$ is $N$, but I don't understand why. I know that the dimension of the lie algebra is $2N^2 + N$, but how does one compute its rank.