Simple roots

Remarks

Example: in $A_2$, take $\alpha = \varepsilon_1 - \varepsilon_2$ and $\beta = \varepsilon_2 - \varepsilon_3$. The roots are $\pm \alpha$, $\pm \beta$ and $\pm (\alpha + \beta)$, since $\varepsilon_1 - \varepsilon_3 = \alpha + \beta$, so $\set{\alpha, \beta}$ is a base. Non-example: $\set{\alpha, \alpha + \beta}$ is a basis of $E$ made of roots, but it fails clause 2, since $\beta = (\alpha + \beta) - \alpha$ has coefficients of both signs. Every root system has a base. For a vector $\gamma \in E$ orthogonal to no root, the positive roots $\beta$ with $(\gamma, \beta) > 0$ that are not a sum of two such roots form a base. For a semisimple Lie algebra, a base is also called a system of simple roots of the Lie algebra.