= F4 root system
{c}
{title2=$\{\pm e_i,\ \pm e_i\pm e_j,\ \tfrac12(\pm e_1\pm e_2\pm e_3\pm e_4)\}$}
In an orthonormal coordinate system on $\mathbb R^4$, these forty-eight vectors form the reduced crystallographic <root system> of type $F_4$. There are twenty-four long roots of squared length two, and twenty-four short roots of squared length one. Signed coordinate permutations preserve the set. Reflections in coordinate roots are sign changes, and reflections in long roots are signed permutations. For the remaining short roots it suffices, by symmetry, to use $h=(1/2,1/2,1/2,1/2)$; its reflection is $v\mapsto v-(\sum_i v_i)(1,1,1,1)/2$. It sends coordinate roots to half-coordinate roots, long roots to long roots, and half-coordinate roots to half-coordinate or coordinate roots. Thus all required reflections preserve the set. The Cartan integrality condition follows directly from its integral or half-integral coordinate pairings.
One <base of a root system> is $e_2-e_3,e_3-e_4,e_4,(e_1-e_2-e_3-e_4)/2$. Its two long roots followed by two short roots give the <F4 Dynkin diagram>, with the double bond between the middle nodes and the arrow toward the short root.
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