Editing Root system (section)

===''F''<sub>4</sub>===
{|  class=wikitable
|+ Simple roots in ''F''<sub>4</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|-
!α<sub>1</sub>
| 1||−1||0||0
|-
!α<sub>2</sub>
|0|| 1||−1||0 
|-
!α<sub>3</sub>
|0||0|| 1||0 
|-
!α<sub>4</sub>
| −{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}}
|- BGCOLOR="#ddd"
|colspan=5 align=center|{{Dynkin2|node_n1|3|node_n2|4b|nodeg_n3|3|nodeg_n4}}
|}
[[File:F4 roots by 24-cell duals.svg|100px|thumb|48-root vectors of F4, defined by vertices of the [[24-cell]] and its dual, viewed in the [[Coxeter plane]]]]
For ''F''<sub>4</sub>, let ''E'' = '''R'''<sup>4</sup>, and let Φ denote the set of vectors α of length 1 or {{radic|2}} such that the coordinates of 2α are all integers and are either all even or all odd. There are 48 roots in this system. One choice of simple roots is: the choice of simple roots given above for ''B''<sub>3</sub>, plus <math display="inline">\boldsymbol\alpha_4 = -\frac{1}{2} \sum_{i=1}^4 e_i</math>.
<!--
\left (
\begin{smallmatrix}
  +1&-1&0&0 \\ 0&+1&-1&0 \\ 0&0&+1&0 \\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}
\end{smallmatrix}
\right )
-->

The ''F''<sub>4</sub> root lattice—that is, the lattice generated by the ''F''<sub>4</sub> root system—is the set of points in '''R'''<sup>4</sup> such that either all the coordinates are [[integer]]s or all the coordinates are [[half-integer]]s (a mixture of integers and half-integers is not allowed).  This lattice is isomorphic to the lattice of [[Hurwitz quaternions]].
{{Clear}}