Editing Root system (section)

===Rank two examples===

In rank 2 there are four possibilities, corresponding to <math>\sigma_\alpha(\beta) = \beta + n\alpha</math>, where <math>n = 0, 1, 2, 3</math>.<ref>{{harvnb|Hall|2015}} Proposition 8.8</ref> The figure at right shows these possibilities, but with some redundancies: <math>A_1\times A_1</math> is isomorphic to <math>D_2</math> and <math>B_2</math> is isomorphic to <math>C_2</math>.

Note that a root system is not determined by the lattice that it generates: <math>A_1 \times A_1</math> and <math>B_2</math> both generate a [[square lattice]] while <math>A_2</math> and <math>G_2</math> both generate a [[hexagonal lattice]].

Whenever Φ is a root system in ''E'', and ''S'' is a [[Linear subspace|subspace]] of ''E'' spanned by Ψ&nbsp;=&nbsp;Φ&nbsp;∩&nbsp;''S'', then&nbsp;Ψ is a root system in&nbsp;''S''.  Thus, the exhaustive list of four root systems of rank&nbsp;2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank.  In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.