Take $L$ a $\mathbb{C}$-Lie algebra with finite dimension and semisimple. Now take $H \subset L$ a maximal toral subalgebra we can define the roots system associated to $(L,H)$ as the set of maps $H\rightarrow\mathbb{C}$ which appear as eigenvalue of $H$ for the adjoint representation.
I would want to prove that the roots system doesn't depend from the choice of $H$.
Thanks in advance for any help