Disclaimer. I am just talking about the chiral part of a conformal field theory. The chiral parts of WZW model are described by affine Kac-Moody algebras. I am describing how to obtain the Kac-Moody algebras starting with abelian currents (derivative of a free bosons). $n$ free Abelian currents $\{J_1,\ldots, J_n\}$ correspond to target space $\mathbb R^n$ and compactifying the target space by a lattice $\Lambda\subset \mathbb R^n$ corresponds to add a vertex operator $V_\alpha$ for every point $\alpha\in \Lambda$ of a lattice (or just for a basis).
By compactifying the target space $\mathbb R^n$ with a root lattice in the case of A-D-E latticea of a simply laced Lie algebra (all roots have the same length $\langle\alpha,\alpha\rangle=2$) one obtains the affine Lie algebra of thethis corresponding Lie groupalgebra at level 1. Namely the $J_n$ and the $V_{\alpha,n}$ with $\alpha\in\Lambda$ and $\langle\alpha,\alpha\rangle=2$ form a basis of the affine Kac-Moody algebra at level 1. The simple simply laced Lie algebras are exactly the one with A,D,E Dynkin diagram. That is basically the Frenkel-Kac construction: I. B. Frenkel and V. G. Kac. Basic representations of affine Lie algebras and dual resonance models. Invent. Math., 62(1):23–66, 1980.
Some other one can get by conformal inclusions, for example $G_2$ at level 3 embeds into $E_6$ at level 1. I guess one obtains all level 1 cases by regarding conformal inclusions $(C_n)_1\times SU(2)_m\subset D_{2m}$ and $G_2\times F_4\subset E_8$.
Finally level $\ell$ one can embed in the $n$-fold tensor product of level 1 by $$J_a^{(\ell)}(z) \sim \sum_k 1 \otimes \cdots \otimes \underbrace{J^{(1)}_a(z)}_{k\text{-th position}}\otimes \cdots\otimes 1$$