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realisation of Kac-Moody Lie algebras
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.