For each root type "A", "B", "C", "D" there are general formulas for calculating a list of positive roots as linear combinations of a given list of simple roots. These linear combinations are given in several texts. See, for example Varadarajan Lie Groups, Lie Algebras and Their Representations, Section 4.4 or Cap and Slovak Parabolic Geometries I. Background and General Theory, Section 2.2.6. Here are these formulas as they are coded into the 3rd calling sequence for the command.

$A_{\mathrm{\𝓁}}$ 

$B_{\mathrm{\𝓁}}$ 

$C_{\mathrm{\𝓁}}$

${\mathrm{D}}_{\mathrm{\𝓁}}$

For any Cartan matrix there is also a simple algorithm for calculating the positive roots from the Cartan matrix. This algorithm is presented in W.A. de Graaf, Lie Algebras: Theory and Algorithms, page 162 and also in W. Fulton and J. Harris, Representation Theory, A First Course, page 330. This approach is coded into the 4th calling sequence for the PositiveRoots command.

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

$\mathrm{PositiveRoots}\left(''A''\,2\right)$

$\mathrm{PositiveRoots}\left(''A''\,3\right)$

$\mathrm{PositiveRoots}\left(''A''\,4\right)$

$\mathrm{PositiveRoots}\left(''A''\,4\right)$

$\mathrm{CM}≔\mathrm{CartanMatrix}\left(''A''\,4\right)$

$\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$

$\mathrm{PositiveRoots}\left(''B''\,2\right)$

$\mathrm{PositiveRoots}\left(''B''\,3\right)$

$\mathrm{PositiveRoots}\left(''B''\,4\right)$

$\mathrm{CM}≔\mathrm{CartanMatrix}\left(''B''\,4\right)$

$\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$

$\mathrm{PositiveRoots}\left(''C''\,2\right)$

$\mathrm{PositiveRoots}\left(''C''\,3\right)$

$\mathrm{PositiveRoots}\left(''C''\,4\right)$

$\mathrm{CM}≔\mathrm{CartanMatrix}\left(''C''\,4\right)$

$\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$

$\mathrm{PositiveRoots}\left(''D''\,2\right)$

$\mathrm{PositiveRoots}\left(''D''\,3\right)$

$\mathrm{PositiveRoots}\left(''D''\,4\right)$

$\mathrm{CM}≔\mathrm{CartanMatrix}\left(''D''\,4\right)$

$\mathrm{PositiveRoots}\left(\mathrm{CM}\right)$