How to compute the norm of a root from the Cartan matrix?

[pscplaton]

Hello!

As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root α from it since its components are invariant under rescaling (if all the simple roots are multiplied by the same constant, the Cartan matrix remains unchanged) ?

The square norm of the root is $α(H_α)$ where $H_α$ is such that $Tr(ad (H_α) ad (h))=α(h)$ for any h in the cartan subalgebra, so it seems to me that there is no freedom for choosing the norm

Thanks!