algebra with Cartan subalgebra and a particular root system . Introducing the notation , one can attempt to extend a Cartan–Weyl basis for to one for the affine Lie algebra, given by , with forming an abelian subalgebra. The eigenvalues of and on are and respectively and independently of . Therefore the root is infinitely degenerate with respect to this abelian subalgebra. Appending the derivation described above to the abelian subalgebra turns the abelian subalgebra into a Cartan subalgebra for the affine Lie algebra, with eigenvalues for Killing form The Killing form can almost be completely determined using its invariance property. Using the notation for the Killing form on and for the Killing form on the affine Kac–Moody algebra, where only the last equation is not fixed by invariance and instead chosen by convention. Notably, the restriction of to the subspace gives a bilinear form with signature . Write the affine root associated with as . Defining , this can be rewritten The full set of roots is Then is unusual as it has zero length: where is the bilinear form on the roots induced by the Killing form. Affine simple root In order to obtain a basis of simple roots for the affine algebra, an extra simple root must be appended, and is given by where is the highest root of , using the usual notion of height of a root. This allows definition of the extended Cartan matrix and extended Dynkin diagrams. Representation theory The representation theory for affine Lie algebras is usually developed using Verma modules. Just as in the case of semi-simple Lie algebras, these are highest weight modules. There are no finite-dimensional representations; this follows from the fact that the null vectors of a finite-dimensional Verma module are necessarily zero; whereas those for the affine Lie algebras are not. Roughly speaking, this follows because the Killing form is Lorentzian in the directions, thus are sometimes called "lightcone coordinates" on the string. The "radially ordered" current operator products can be understood to be time-like normal ordered by taking with the time-like direction along the string world sheet and the spatial direction. Vacuum representation of rank k The representations are constructed in more detail as follows. Fix a Lie algebra and basis . Then is a basis for the corresponding loop algebra, and is a basis for the affine Lie algebra . The vacuum representation of rank