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Let $R$ be a root system on a Euclidean space $\mathbb{E}$ with simple roots $\{\alpha_i : i=1,....,l\}$. If $\alpha$ = $\sum_{i=1}^{l}c_i\alpha_i$ be a root , then show that $\frac {c_i(\alpha_i|\alpha_i)}{(\alpha|\alpha)}$ $\in$ $\mathbb{Z}$ $\forall$ $i$ = $1, 2....,l$. Any help will be highly appreciated.

Ester
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    Could you elaborate on which inner product you denote by $(\cdot\mid\cdot)$? There tend to be about a million different ones floating around when dealing with root systems, and which ones have which properties (and why) depend on which one it is. – Tobias Kildetoft Jun 07 '16 at 10:55
  • (.|.) is a symmetric bilinear form on E which is invariant under the Weyl Group. – Ester Jun 07 '16 at 12:12

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