root system
When you diagonalize a semisimple Lie algebra against a Cartan subalgebra, the off-diagonal directions sort themselves into a finite constellation of vectors in a Euclidean space — the roots. Remarkably these vectors form a rigid, highly symmetric pattern: reflect any root across the hyperplane perpendicular to another and you land back on a root. A root system distills all this rigidity into pure combinatorial geometry, divorced from the Lie algebra that produced it.
Abstractly, a root system is a finite set R of nonzero vectors spanning a Euclidean space V such that: (1) for each root alpha, the only multiples of alpha in R are alpha and -alpha; (2) the reflection s_alpha across the hyperplane perpendicular to alpha permutes R; and (3) the integrality condition holds, that 2(alpha . beta)/(beta . beta) is an integer for all roots alpha, beta. The reflections generate the finite Weyl group.
Choosing a set of positive roots picks out a basis of simple roots; every root is then an integer combination of simple roots with all coefficients of one sign. The integrality condition severely constrains the possible angles between simple roots (only 90, 120, 135, 150 degrees occur), and this is exactly what makes irreducible root systems classifiable: they fall into the four infinite families A_n, B_n, C_n, D_n and five exceptionals G_2, F_4, E_6, E_7, E_8.
The root system A_2 (from sl(3)) has six roots forming a regular hexagon: two simple roots alpha and beta at 120 degrees, plus alpha + beta, and the three negatives. The Weyl group is S_3, the symmetries permuting the three pairs of opposite roots.
A_2: six roots in a regular hexagon, Weyl group S_3.
Root systems are the bridge from algebra to combinatorics: classifying simple complex Lie algebras reduces to classifying irreducible root systems, which reduces to classifying connected Dynkin diagrams — a purely graph-theoretic problem.