Simple root (${\alpha }_i$)

The simple roots are the positive roots that cannot be expressed as sums of two positive roots. Every root is an integer combination of simple roots with coefficients of one sign.

If a positive root $\alpha$ is not simple, then $\alpha -\beta$ is a positive root for some simple root $\beta$. Iteration writes every positive root as a sum of simple roots whose successive partial sums are roots.

For a simple root $\alpha$, the reflection $s_{\alpha }$ sends $\alpha$ to $-\alpha$ and permutes all other positive roots.