Principal basis in Cartan subalgebra

Let $\text{Lie}g$ be a simple complex Lie algebra and $\text{Lie}h$ a Cartan subalgebra. In this article we explain how to obtain the principal basis of $\text{Lie}h$ starting form a set of generators $\{p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_r\}$,$r=\text{rank}(\text{Lie}g)$, of the invariants polynomials $\text{Sgg}$. For each invariant polynomial $p$, we define a $G$-equivariant map $Dp$ form $\text{Lie}g$ to $\text{Lie}g$. We show that the Gram-Schmidt orthogonalization of the elements $\{D{p}_1({\rho }^{\vee }),\mathrm{.}\mathrm{.}\mathrm{.}D{p}_r({\rho }^{\vee })\}$ gives the principal basis of $\text{Lie}h$. Similarly the orthogonalization of the elements ${Dp_1(\rho),

… Dp_r(\rho) }$producestheprincipalbasisoftheCartansubalgebraof$\lLie{g}$,theLanglandsdualof$\Lie{g}$.