I am reading about infinite dimensional lie algebras from the books Infinite-Dimensional Lie Algebras by Victor G. Kac and Lie Algebras of Finite and Affine Type by Roger Carter. Now I have gone through the basic construction rules of realaisation of a matrix $A$ and then constructing the auxiliary lie algebra $\hat {\mathfrak g}(A)$ and then finally the required algebra $\mathfrak g(A)$ which is called the Kac-Moody Algebra if $A$ is a Generalised Cartan Matrix (GCM).
Now I am not finding a general example of this construction. Please help me.
I have found an example for the symmetric $n \times n$ matrix $A$ over $\mathbb C$ with $A_{i,j} = 2 \forall i=j \text{ , } A_{i,i+1}=-1$ and $0$ in all other places (This is the Cartan Matrix for the graph type $A_n$). The algebra $\mathfrak g(A)$ here is nothing but the lie algebra $\mathfrak {sl}_{n+1}(\mathbb C)$.
But I am not finding a suitable example where rank of the matrix is strictly less than the order of a the matrix (Note that in the above matrix $A$ has full rank).
It will be very helpful for me if someone kindly show me such a generalised example. I can then understand the construction more clearly. If such an example can be found in any book or paper please give me the details. I will definitely have a look.
Please help me. Thnx in advance.