Editing Root system (section)

===''B''<sub>''n''</sub>===
{|  class=wikitable
|+ Simple roots in ''B''<sub>4</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|-
!α<sub>1</sub>
| &nbsp;1||−1||0||0
|-
!α<sub>2</sub>
|0|| &nbsp;&nbsp;1||−1||0 
|-
!α<sub>3</sub>
|0||0|| &nbsp;1||−1 
|-
!α<sub>4</sub>
|0||0|| 0||&nbsp;1
|- BGCOLOR="#ddd"
|colspan=5 align=center|{{Dynkin2|node_n1|3|node_n2|3|node_n3|4b|nodeg_n4}}
|}
Let ''E'' = '''R'''<sup>''n''</sup>, and let Φ consist of all integer vectors in ''E'' of length 1 or {{radic|2}}.  The total number of roots is 2''n''<sup>2</sup>.  One choice of simple roots is {{math|1='''α'''<sub>''i''</sub> = '''e'''<sub>''i''</sub> – '''e'''<sub>''i''+1</sub>}} for {{math|1 ≤ ''i'' ≤ ''n'' – 1}} (the above choice of simple roots for ''A''<sub>''n''−1</sub>), and the shorter root {{math|1='''α'''<sub>''n''</sub> = '''e'''<sub>''n''</sub>}}.

The reflection ''σ''<sub>''n''</sub> through the hyperplane perpendicular to the short root '''α'''<sub>''n''</sub> is of course simply negation of the ''n''th coordinate. 
For the long simple root '''α'''<sub>''n''−1</sub>, σ<sub>''n''−1</sub>('''α'''<sub>''n''</sub>) = '''α'''<sub>''n''</sub> + '''α'''<sub>''n''−1</sub>, but for reflection perpendicular to the short root, ''σ''<sub>''n''</sub>('''α'''<sub>''n''−1</sub>) = '''α'''<sub>''n''−1</sub> + 2'''α'''<sub>''n''</sub>, a difference by a multiple of 2 instead of 1.

The ''B''<sub>''n''</sub> root lattice—that is, the lattice generated by the ''B''<sub>''n''</sub> roots—consists of all integer vectors.

''B''<sub>1</sub> is isomorphic to ''A''<sub>1</sub> via scaling by {{radic|2}}, and is therefore not a distinct root system.
{{Clear}}