What is the type $B_2$ Cartan matrix?

Question

Is type $B_2$ Cartan matrix $\left( \begin{matrix} 2 & -1 \\ -2 & 2 \end{matrix} \right)$ or $\left( \begin{matrix} 2 & -2 \\ -1 & 2 \end{matrix} \right)$? It seems that different books and papers use different $B_2$ Cartan matrix. Thank you very much.

Answer 1

What's the difference? If $r_1$ and $r_2$ are simple roots of the simple Lie algebra corresponding to $B_2$ and if$$\begin{bmatrix}2\frac{(r_1,r_1)}{(r_1,r_1)}&2\frac{(r_1,r_2)}{(r_1,r_1)}\\2\frac{(r_2,r_1)}{(r_2,r_2)}&2\frac{(r_2,r_2)}{(r_2,r_2)}\end{bmatrix}=\begin{bmatrix}2&-1\\-2&2\end{bmatrix},$$then$$\begin{bmatrix}2\frac{(r_2,r_2)}{(r_2,r_2)}&2\frac{(r_2,r_1)}{(r_2,r_2)}\\2\frac{(r_1,r_2)}{(r_1,r_1)}&2\frac{(r_1,r_1)}{(r_1,r_1)}\end{bmatrix}=\begin{bmatrix}2&-2\\-1&2\end{bmatrix}.$$