Let ${\mathrm{\Δ}}_{ }\subseteq {\mathrm{\ℂ}}^m$be a list of roots for either an abstract root system or for a simple Lie algebra. In particular, ${\mathrm{\Δ}}_{ }$must have an even number of elements and if $X \in {\mathrm{\Δ}}_{}\,$then $-X \in {\mathrm{\Δ}}_{}$. Write $\mathrm{\Δ} \= {\mathrm{\Δ}}_{ }^{\+}\bigcup {\mathrm{\Δ}}^{-}$ where, if $X \in {\mathrm{\Delta }}_{}^{\+}$then $-X \in {\mathrm{\Δ}}^{-}$and $X \in {\mathrm{\Delta }}_{}^{-}$then $-X \in {\mathrm{\Δ}}^{\+}\.$The set ${\mathrm{\Δ}}^{\+}$is called the set of positive roots .The choice of positive roots is not unique. If ${\mathrm{\Delta }}^{\+}$ is set of positive roots,then a root ${\mathrm{\α}}_{} \in {\mathrm{\Δ}}^{\+}$is called a simple root if it is not a sum of any other 2 positive roots. If ${\mathrm{\Δ}}_0$is a set of simple roots for ${\mathrm{\Δ}}^{\+}$, then every root in ${\mathrm{\Δ}}^{\+}$is a linear combination of the roots in ${\mathrm{\Δ}}_0$ with positive integer coefficients.The number of simple roots equals the rank of the Lie algebra.

The command SimpleRoots(PR) returns a list of vectors defining a set of simple roots.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left(''sp(8,\; R))''\,\mathrm{sp8R}\,\mathrm{labelformat}=''gl''\,\mathrm{labels}=\left['E'\,'\mathrm{\omega }'\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; sp8R}$

$\mathrm{CSA\_sp8R}≔\left[\mathrm{E11}\,\mathrm{E22}\,\mathrm{E33}\,\mathrm{E44}\right]$

$\mathrm{CSA\_sp8R}:=\left[\mathrm{E11}\,\mathrm{E22}\,\mathrm{E33}\,\mathrm{E44}\right]$

$\mathrm{RSD\_sp8R}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA\_sp8R}\right)$

$\mathrm{RSD\_sp8R}:=\mathrm{table}\left(\left[\left[1\,0\,1\,0\right]\=\mathrm{E17}\,\left[0\,-1\,0\,1\right]\=\mathrm{E42}\,\left[0\,0\,1\,1\right]\=\mathrm{E38}\,\left[1\,0\,0\,-1\right]\=\mathrm{E14}\,\left[-2\,0\,0\,0\right]\=\mathrm{E51}\,\left[0\,0\,0\,2\right]\=\mathrm{E48}\,\left[0\,0\,0\,-2\right]\=\mathrm{E84}\,\left[0\,0\,2\,0\right]\=\mathrm{E37}\,\left[0\,1\,1\,0\right]\=\mathrm{E27}\,\left[0\,0\,1\,-1\right]\=\mathrm{E34}\,\left[-1\,0\,1\,0\right]\=\mathrm{E31}\,\left[0\,0\,-2\,0\right]\=\mathrm{E73}\,\left[0\,-2\,0\,0\right]\=\mathrm{E62}\,\left[-1\,0\,0\,-1\right]\=\mathrm{E54}\,\left[0\,-1\,1\,0\right]\=\mathrm{E32}\,\left[0\,1\,0\,-1\right]\=\mathrm{E24}\,\left[0\,0\,-1\,-1\right]\=\mathrm{E74}\,\left[1\,0\,0\,1\right]\=\mathrm{E18}\,\left[0\,1\,-1\,0\right]\=\mathrm{E23}\,\left[2\,0\,0\,0\right]\=\mathrm{E15}\,\left[1\,0\,-1\,0\right]\=\mathrm{E13}\,\left[0\,-1\,0\,-1\right]\=\mathrm{E64}\,\left[0\,2\,0\,0\right]\=\mathrm{E26}\,\left[1\,-1\,0\,0\right]\=\mathrm{E12}\,\left[1\,1\,0\,0\right]\=\mathrm{E16}\,\left[0\,1\,0\,1\right]\=\mathrm{E28}\,\left[-1\,-1\,0\,0\right]\=\mathrm{E52}\,\left[0\,-1\,-1\,0\right]\=\mathrm{E63}\,\left[-1\,1\,0\,0\right]\=\mathrm{E21}\,\left[-1\,0\,-1\,0\right]\=\mathrm{E53}\,\left[-1\,0\,0\,1\right]\=\mathrm{E41}\,\left[0\,0\,-1\,1\right]\=\mathrm{E43}\right]\right)$

$\mathrm{PR\_sp8R}≔\mathrm{PositiveRoots}\left(\mathrm{RSD\_sp8R}\,⟨1\,2\,3\,4⟩\right)$

$\mathrm{SR\_sp8R}≔\mathrm{SimpleRoots}\left(\mathrm{PR\_sp8R}\right)$

$\mathrm{GetComponents}\left(\mathrm{PR\_sp8R}\,\mathrm{SR\_sp8R}\right)$

$\left[\left[1\,1\,1\,0\right]\,\left[1\,0\,0\,1\right]\,\left[2\,1\,2\,1\right]\,\left[2\,1\,2\,2\right]\,\left[2\,1\,2\,0\right]\,\left[1\,1\,2\,0\right]\,\left[1\,0\,1\,0\right]\,\left[1\,0\,0\,0\right]\,\left[1\,1\,1\,1\right]\,\left[0\,1\,0\,0\right]\,\left[0\,1\,2\,0\right]\,\left[0\,1\,1\,0\right]\,\left[1\,1\,2\,1\right]\,\left[0\,0\,1\,0\right]\,\left[1\,0\,1\,1\right]\,\left[0\,0\,0\,1\right]\right]$