Representations of contragredient Lie algebras in contact vector fields (Q810623)

Let \(K_ 5\) be the Lie algebra of contact vector fields \[ v_ f=- \frac{\partial f}{\partial p}\frac{\partial}{\partial x}-\frac{\partial f}{\partial q}\frac{\partial}{\partial y}+\left(f-p\frac{\partial f}{\partial p}-q\frac{\partial f}{\partial q}\right)\frac{\partial}{\partial u}+ \left(\frac{\partial f}{\partial x}+p\frac{\partial f}{\partial u}\right)\frac{\partial}{\partial p}+\left(\frac{\partial f}{\partial y}+q\frac{\partial f}{\partial u}\right)\frac{\partial}{\partial q}. \] The authors give the list of all simple Lie subalgebras \(L\subset K_ 5\). There are the following Kac-Moody subalgebras in \(K_ 5:\) \(A_ 1^{(1)}\), \(A_ 1^{(2)}\), \(A_ 2^{(1)}\), \(C_ 2^{(1)}\). The authors state without proof that every simple Lie subalgebra or Kac-Moody algebra in \(K_{2n-1}\) satisfies \(\text{rank}\, L\leq n\).