Editing Root system (section)

===''D''<sub>''n''</sub>===
{|  class=wikitable
|+ Simple roots in ''D''<sub>4</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|- valign=top
!α<sub>1</sub>
| &nbsp;1||−1||0||0
|-
!α<sub>2</sub>
|0|| &nbsp;1||−1||0 
|-
!α<sub>3</sub>
|0||0|| &nbsp;1||−1 
|-
!α<sub>4</sub>
|0||0|| &nbsp;1||&nbsp;1
|- BGCOLOR="#ddd"
|colspan=5 align=center|[[File:DynkinD4 labeled.png|80px]]<!--{{Dynkin2|node_n1|3|branch|3|node_n3}}-->
|}
Let {{math|1=''E'' = '''R'''<sup>''n''</sup>}}, and let Φ consist of all integer vectors in ''E'' of length {{radic|2}}.  The total number of roots is {{math|2''n''(''n'' − 1)}}. 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 {{math|''A''<sub>''n''−1</sub>}}) together with {{math|1='''α'''<sub>''n''</sub> = '''e'''<sub>''n''−1</sub> + '''e'''<sub>''n''</sub>}}.

Reflection through the hyperplane perpendicular to '''α'''<sub>''n''</sub> is the same as [[Transposition (mathematics)|transposing]] and negating the adjacent ''n''-th and (''n'' − 1)-th coordinates. Any simple root and its reflection perpendicular to another simple root differ by a multiple of 0 or 1 of the second root, not by any greater multiple.

The ''D''<sub>''n''</sub> root lattice – that is, the lattice generated by the ''D''<sub>''n''</sub> roots – consists of all integer vectors whose components sum to an even integer.  This is the same as the ''C''<sub>''n''</sub> root lattice.

The ''D''<sub>''n''</sub> roots are expressed as the vertices of a [[Rectification_(geometry) | rectified]] ''n''-[[orthoplex]], [[Coxeter–Dynkin diagram]]: {{CDD|node|3|node_1|3}}...{{CDD|3|node|split1|nodes}}. The {{math|2''n''(''n'' − 1)}} vertices exist in the middle of the edges of the ''n''-orthoplex.

''D''<sub>3</sub> coincides with ''A''<sub>3</sub>, and is therefore not a distinct root system. The twelve ''D''<sub>3</sub> root vectors are expressed as the vertices of {{CDD|node|split1|nodes_11}}, a lower symmetry construction of the [[cuboctahedron]].

''D''<sub>4</sub> has additional symmetry called [[triality]]. The twenty-four ''D''<sub>4</sub> root vectors are expressed as the vertices of {{CDD|node|3|node_1|split1|nodes}}, a lower symmetry construction of the [[24-cell]].
{{Clear}}