Root system

A (reduced) root system is a finite subset $\Phi\subset V\setminus\{0\}$ of a real finite-dimensional inner product space $(V,\langle\cdot,\cdot\rangle)$ such that:

1. $\Phi$ spans $V$.
2. If $\alpha\in\Phi$, then the only scalar multiples of $\alpha$ in $\Phi$ are $\pm\alpha$.
3. For each $\alpha\in\Phi$, the reflection
   $$s_\alpha(v)=v-\frac{2\langle v,\alpha\rangle}{\langle \alpha,\alpha\rangle}\,\alpha$$
   preserves $\Phi$, i.e. $s_\alpha(\Phi)=\Phi$.
4. (Integrality) For all $\alpha,\beta\in\Phi$,
   $$\frac{2\langle \beta,\alpha\rangle}{\langle \alpha,\alpha\rangle}\in\mathbb Z.$$

The subgroup of $\mathrm{GL}(V)$ generated by the reflections $\{s_\alpha\}$ is the Weyl group of $\Phi$.

For a complex semisimple Lie algebra $\mathfrak g$ with Cartan subalgebra $\mathfrak h$, the roots lie in $\mathfrak h^*$. Their real span is a Euclidean space under the positive-definite form induced from a compact real form (equivalently, from a suitable real form of the Killing form), and the roots form a reduced root system there. The corresponding root-space decomposition is the bridge between the Lie algebra and this combinatorial data.

Choosing a positive system picks out a basis of simple roots, from which one constructs the Cartan matrix and Dynkin diagram. This data underlies the classification of simple Lie algebras.