According to P. Francesco et al. conformal field theory book the conformal charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which obtained by Sugawara construction on the generators of symmetry in the Wess-Zumino-Witten (WZW) model, is given by the following formula (Page 629 eq 15.61):

$$c=\frac{k\dim g}{k+h^\lor }$$ Where $\dim g$ is the dimension of the corresponding algebra $g$, $k$ is the level of the affine lie algebra $\hat{g}_k$ and $h^\lor$ is the dual Coxeter number of $g$.

My question is concerning the specific Lie algebra $g = sl(2,\mathbb{R})$. This Lie algebra is noncompact but has finite dimensions of $\dim sl(2,\mathbb{R})=3$, but I am concerned with finding what is value of the dual Coxeter number. According to the classification of semisimple Lie algebra visualized by Dynkin diagrams, the closest group is of the class of $A_n$ corresponding to $sl(n+1,\mathbb{C})$ with the dual Coxeter number $h^{\lor}=n+1$ according to Wikipedia but in our case, $n=1$ but the group is defined over $\mathbb{R}$ and not over $\mathbb{C}$. According to Maldecena et al. Article the result which appears in eq. 65 demanding the dual Coxeter number to have the following value: $$h^\lor=-2$$ This value is different than the complex case of $sl(2,\mathbb{C})$ what is the reason for this? moreover, what is the right class $(A_n,B_n,C_n,D_n,...)$ that is suitable for $sl(2,\mathbb{R})$ which will give eventually the right answer?

According to P. Francesco et al. conformal field theory book the central charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which obtained by Sugawara construction on the generators of symmetry in the Wess-Zumino-Witten (WZW) model, is given by the following formula (Page 629 eq 15.61):

$$c=\frac{k\dim g}{k+h^\lor }$$ Where $\dim g$ is the dimension of the corresponding algebra $g$, $k$ is the level of the affine lie algebra $\hat{g}_k$ and $h^\lor$ is the dual Coxeter number of $g$.

My question is concerning the specific Lie algebra $g = sl(2,\mathbb{R})$. This Lie algebra is noncompact but has finite dimensions of $\dim sl(2,\mathbb{R})=3$, but I am concerned with finding what is value of the dual Coxeter number. According to the classification of semisimple Lie algebra visualized by Dynkin diagrams, the closest group is of the class of $A_n$ corresponding to $sl(n+1,\mathbb{C})$ with the dual Coxeter number $h^{\lor}=n+1$ according to Wikipedia but in our case, $n=1$ but the group is defined over $\mathbb{R}$ and not over $\mathbb{C}$. According to Maldecena et al. Article the result which appears in eq. 65 demanding the dual Coxeter number to have the following value: $$h^\lor=-2$$ This value is different than the complex case of $sl(2,\mathbb{C})$ what is the reason for this? moreover, what is the right class $(A_n,B_n,C_n,D_n,...)$ that is suitable for $sl(2,\mathbb{R})$ which will give eventually the right answer?