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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 [itex]α(H_α)[/itex] where [itex]H_α[/itex] is such that [itex]Tr(ad (H_α) ad (h))=α(h)[/itex] for any h in the cartan subalgebra, so it seems to me that there is no freedom for choosing the norm

Thanks!