realisation of Kac-Moody Lie algebras

2018-08-25 11:55:59

I am reading Infinite dimensional lie algebras by Kac. We start with a $n \times n$ GCM $A$ of rank $l$, then he defines the realization associated with the matrix $A$ which is of dimension $2n-l$. I know that in the simple Lie algebra case this dimension is $n$ as

I don't understand why we are taking the space of dimension $2n-l$?

Please explain, When the GCM is not of finite type, what we are getting extra by this definition of Lie algebra?

Thanks for your time.