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Given a universal elementary Chevalley-Demazure group $E_\Phi^{sc}(R)$ forwhich its (standard) parabolic subgroups are finitely generated, we considerthe problem of classifying which parabolics $P(R) \subset E_\Phi^{sc}(R)$ arefinitely presented. We show that, under mild assumptions, this is equivalent tothe finite presentability of a suitable retract of $P$ which contains the Levifactor. If the base ring $R$ is a Dedekind domain of arithmetic type, wecombine our results with well-known theorems due to Borel-Serre, Abels, Behrand Bux to give a partial classification of finitely presentable $S$-arithmeticsubgroups of parabolics in split reductive linear algebraic groups.

Presentations of parabolics in some elementary Chevalley-Demazure groups