What is the Cartan matrix of type $A_1 \times A_1$?

Question

The Cartan matrix of type $A_2$ is $\left( \begin{matrix} 2 & -1 \\ -1 & 2 \end{matrix} \right)$. What is the Cartan matrix of type $A_1 \times A_1$? Thank you very much.

Answer 1

It's $$ \left( \begin{matrix} 2 & 0 \\ 0 & 2 \end{matrix} \right). $$

Recall the formula for the entries of the Cartan matrix, $a_{ij}=2(\alpha_i,\alpha_j)/(\alpha_i,\alpha_i)$. In $A_1\times A_1$, the two simple roots are orthogonal (the corresponding vertices in the Dynkin diagram have no edge), so the inner product in the numerator is zero for the off diagonal entries.