Dual root system (${\Phi }^{\vee }$)

The dual root system of $\Phi$ is ${\Phi }^{\vee }=\{{\alpha }^{\vee }:\alpha \in \Phi \}$. It has the transposed Cartan matrix, so its Dynkin diagram reverses every multiple-edge arrow; in particular, $B_n$ and $C_n$ are dual.