Positive root

A choice of positive roots selects exactly one of $\alpha$ and $-\alpha$ and is closed under addition whenever the sum is a root.

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.

The highest root is the maximal positive root in the root order determined by the simple roots.

The Weyl vector is $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$. It satisfies $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ for every simple root.