It is known that $B_2$ is isomorphic to $C_2$; by looking at their Dynkin diagrams, it looks like one can be obtained from the other through a rotation of $\pi$. Also, if $B$ is the Cartan matrix of $B_2$ and $C$ is the Cartan matrix of $C_2$, $B^{t} = C$. Is this true in general (i.e., rotation of $\pi$ of Dynkin diagrams corresponds to transposition of their respective Cartan matrices)?
What about the isomorphism between $D_3$ and $A_3$? Clearly the construction of Cartan matrix is strictly dependent on the order established between roots in the Dynkin diagram: is there an operation on the matrices that allows us to express the isomorphism for this case as well?