= Solution
A reduced crystallographic abstract <root system> in a finite-dimensional real <inner product space> $E$ is a finite set $R$ such that: it spans $E$ and does not contain zero; the only scalar multiples of $\alpha\in R$ belonging to $R$ are $\pm\alpha$; every <root reflection>
$$
s_\alpha(x)=x-\frac{2(x,\alpha)}{(\alpha,\alpha)}\alpha
$$
permutes $R$; and each <Cartan integer> $2(\beta,\alpha)/(\alpha,\alpha)$ is integral. These are the crystallographic and reduced conventions appropriate to <roots> of a complex <semisimple Lie algebra>. Without crystallographic integrality, the numerical restrictions in the following subparts would not hold.
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