I am studying simple Lie algebras and, on page 129 of the book "Group Theory A physicist's Survey", by Pierre Ramond, it is claimed at the bottom of the page that << Hence the difference of two simple roots must be zero >>.
To arrive at this statement the author recalls that for positive roots to be simple as well << (...) none can be expressed as linear combination of other simple roots with positive coeficients.>> These simple roots are introduced under the notation: $\alpha_i, i =1, 2, ..., r$ where r is the rank of the algebra.
MY QUESTION follows from the equalities at the end of the same page that should allow to conclude that the difference between two simple roots is zero.
Using the difference between two simple roots, $\alpha_{\Delta} = \alpha_i - \alpha_j$, to write $\alpha_i$ as
$\alpha_i = \alpha_{\Delta} + \alpha_j$
I can only immediately conclue that $\alpha_{\Delta}$ is not another simple root otherwise it would violate the given definition. But how can I conlude that it must be zero? If such statment is true, the previous equation becomes $\alpha_i = \alpha_j$ which is not looking good.
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