For three complete flags in R^d the exact worst-case simultaneous generating set has size floor(5d/3), and the paper determines the exact value for any number of flags.
Bourbaki, Groupes et Alg` ebres de Lie, Chapitres 4 ` a 6
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Simultaneous generating sets for flags
For three complete flags in R^d the exact worst-case simultaneous generating set has size floor(5d/3), and the paper determines the exact value for any number of flags.